\section{Introduction}
Many existing multimedia annotation and artificial intelligence  systems benefit from large-scale datasets with accurate annotations. For example, $40$ participants were gathered to rate the attractiveness scores for $1240$ subjects in Beauty Sense project~\cite{BeautySense,TOMM}. In another work, $69$ participants  were requested to provide touch saliency for at least $400$ images~\cite{Mengdi}. However, constructing such datasets is generally tedious and labor-intensive, as it involves massive manual-cropping and hand-labeling. Therefore, games with a purpose (GWAPs) are a simple yet powerful means to collect useful information from players in a way that is entertaining for them. A GWAP embeds  data collection implicitly into the game playing process. It is developed from the human computation technique which relies on the collaboration of game players to solve problems. Each player performs a small part of a massive computation, similar to a processor in a distributed system. Unlike computer processors, however, humans require a certain incentive to participate in such collective computation. GWAP aims to implicitly embedding the computational tasks in an enjoyable and attractive games while guaranteeing correct solutions to the problems.

Over the past few years, many GWAPs~\cite{ahnlab,ahngam,labelme,phog} have been proposed to exploit the brainpower made available by multitudes of casual gamers to perform tasks that, despite being relatively easy for humans to complete, are difficult for machines. GWAPs are best known for image annotation. In the `ESP' game \cite{ahnlab}, players guess content objects or properties of random images by typing what they see when it appears on the screen. Other image annotation games include: Matchin \cite{hacmat}, a two-player game in which each player is shown two images and asked to click on the image their partner would prefer, and Phetch \cite{ahnimp}, a game that collects explanatory descriptions of images in order to improve Web accessibility for the visually impaired.  Recently,  Purposive Hidden-Object-Game~\cite{phog} was proposed. It was designed to imperceptibly embed localizing objects into an enjoyable game process thus attracting many people to make voluntary contribution to annotating images.

Besides images, GWAPs have been used for video annotation. For example, OntoTube \cite{sioont}, Yahoo's Videotaggame \cite{zwovid}, and Waisd \cite{add100}, are all games in which two players have to quickly agree on a set of tags for the same streaming YouTube video. GWAPs have also been exploited to automatically tag music tracks with semantic labels. HerdIt \cite{baruse}, for example, asks players to accomplish various tasks and answer quizzes related to the song they are listening to. In Tagatune \cite{lawtag}, two players listen to an audio file and describe to the other what each is hearing. Players must then decide whether or not the game has played the same soundtrack to both participants.

One of the most interesting tasks GWAPs can be used for is commonsense knowledge acquisition from members of the general public \cite{chklea,speope,camacq}. One example, Verbosity \cite{ahnver}, is a real time quiz game for collecting commonsense facts. In the game, two players take different roles at different times: one functions as a narrator, who has to describe a word using templates, while the other has to guess the word in the shortest time possible.
Another example is FACTory Game \cite{lencyc}, a GWAP developed by Cycorp which randomly chooses some facts and presents them to players in order for them to guess whether a statement is true or false. A variant of the FACTory game is the Concept Game on Facebook \cite{herthe}, which collects commonsense knowledge by proposing random assertions to users and asks them whether the given assertion is meaningful or not.

Questions are very important in collecting commonsense. Kuo and Hsu~\cite{Kuo} introduced the problem of resource-bounded crowdsourcing of commonsense knowledge and proposed an approach to find a productive question set automatically. Another problem with existing GWAPs is that the information gathered from them is often unrecyclable; acquired data is often applicable only to the specific stimuli encountered during gameplay.  Moreover, such games often have a fairly low `sticky factor', and are often unable to engage gamers for more than a couple of minutes. The fun factor sometimes is not considered during the design of such a GWAP system.

In this paper, we propose a 3D game engine for commonsense knowledge acquisition (GECKA3D), that aims to overcome the main drawbacks of traditional data-collecting games by empowering users to create their own GWAPs and by mining knowledge that is highly reusable and multi-purpose. In addition, we also consider adding game mechanism to enhance the fun factor to engage the players. In particular, the GECKA3D framework has two modes, editor and player. The editor mode allows game designers to create compelling serious games for their peers to play. In doing so, they gather commonsense knowledge useful for intelligent applications in any field requiring in-depth knowledge of the real world, including reasoning, perception and social systems simulation \cite{camcomshort}. Meanwhile, the player mode is used by players to perform the task. The framework was built on top of the cocos2d game engine\footnote{http://www.cocos2d-x.org}. Note that the cocos2d game engine is a suite of open-source, cross-platform, and game-development tools used by thousands of developers all over the world.

\section{Proposed Framework}
An important difference between traditional artificial intelligence systems and human intelligence is the human ability to harness commonsense knowledge gathered from a lifetime of learning and experience to make informed decisions~\cite{camacq}. This allows humans to adapt easily to novel situations where artificial intelligence fails catastrophically due to a lack of situation-specific rules and generalization capabilities. Commonsense knowledge also provides background information enabling humans to successfully operate in social situations where such knowledge is typically assumed \cite{camaf2}. To this end, GECKA3D allows users to design compelling serious games via editor mode. As opposed to traditional GWAPs, GECKA3D does not limit users to specific, often tedious, tasks, but rather gives them the freedom to choose both the kind and the granularity of knowledge to be encoded, through a user-friendly and intuitive interface. Not just a system for the creation of microgames, GECKA3D is a serious game engine that aims to give designers the means to create long adventure games to be played by others. GECKA3D offers functionalities typical of role-play games (RPGs), e.g., a question/answer dialogue box enabling communication and the exchange of objects (optionally tied to correct answers) between players and virtual world inhabitants, a library for enriching scenes with useful and yet visually-appealing objects, backgrounds, characters.

\makefigure{t}{gecka3dscene1}{Editor mode. Game designers can drag and drop objects and characters from the library and specify how these interact with each other.}

\subsection{Initial Knowledge Construction}
The core elements in the proposed GECKA3D framework are objects and actions. Their semantics can be specified through prerequisite-object-action-goal (POAG) quartets. Prerequisites indicate what must be present or have been done before applying the object or action. Objects indicate states or objects of the world (including emotional states, e.g., ``if I give food to hungry people, their happiness is likely to rise''). Goals in turn specify the specific scene goals that are facilitated by that particular POAG quartet. In order to provide designers with initial knowledge, we first build the corpus of Object-Action groups. We consider verbs as actions whereas nouns can be regarded as objects. In the initial step of corpus retrieval, we utilized the combinations of verbs and nouns as the search terms fed into Microsoft Bing search engine\footnote{http://www.bing.com} in the form of `how to + a $<$noun$>$'. The number of searched results can be considered as the joint likelihood of the combination of action and object. We also integrate boost library\footnote{http://www.boost.org} for semantic parsing. In the post-processing step, we completed cleaning of verb and noun corpuses with normalization, e.g., forcing all characters to lowercase, and removing html tags. As a result, we collect the initial corpus with 1500 nouns and 636 verbs in English.

\subsection{Editor Mode}
Unlike other serious games or data collection tools~\cite{ahnpee,labelme,phog,TamMM13}, GECKA3D provides authoring tools (or so called editor mode) for designers to create the training scenarios. The editor mode is built in 2.5D which is similar as 2D graphical projections and simulates the appearance of being three-dimensional (3D). In other words, the gameplay is in a 3D video game that is restricted to a 2D plane or has a virtual camera with a fixed angle.

This is inspired by games such as SimCity\footnote{https://en.wikipedia.org/wiki/SimCity}. Figure~\ref{fig:gecka3dscene1}  illustrates the editor mode in being used by a designer.
The designers are able to use the editor to create the training scenario in the form of tile based levels. The GECKA3D framework allows designers to define actions, objects, and goals. In case an action or an object is not available in the library, GECKA3D allows game designers to define their own custom items by building shapes from a set of predefined geometric forms or applying transformations to existing items. This enables the creation of new objects for which there is no available icon by combining available graphics and predefined shapes, and the use of transformations to create various object states, such as a `broken cup'. The ability of users to create their own custom items and actions is key to maintaining an undisrupted game flow.

Each game has enter/exit portal sprites. The designer has to define the starting location of the in-game character. It is easy for designer to know which portal when defining entrance. This feature allows the character to traverse in different levels. As mentioned in~\cite{ahngam}, people play games not because they are personally interested in solving an instance of a computational problem but because they wish to be entertained. Therefore, in order to gain the sticky factor of the game, GECKA3D embeds some game mechanism. For example, it allows designers to add monsters/zombies to chase the in-game player. Therefore, the player has to make decision quickly in order to avoid the pursuers.

\makefigure{!t}{gecka3dprerequisites}{The designers specify the item combination rule. They also define the outcome of the combination. Note that items can be objects or actions.}

\subsection{Object-Action Combinations
Game designers drag and drop objects and characters from action/object libraries into scenes. In addition, the designers can add item combinations and the corresponding outcomes, for example, orange + blender = orange juice. In particular, designers can create, load and delete such item combinations. They can specify POAG quartet by grouping objects with the actions performed over it (Fig.~\ref{fig:gecka3dprerequisites}). POAG quartets give us pieces of commonsense information like ``the result of blending an orange with a blender, is orange juice''.

Towards the goal of improving gameplay, and since we are mainly interested in typical commonsense knowledge, POAG quartets associated with a specific object type are shared among all the instances of such an object (`inheritance'). Whenever a game designer associates a POAG to an object in the scene, that POAG instantly becomes shared among all the other objects of the same type, no matter if these are located in different scenes. New instances inherit this POAG as well. Game designers, however, can create exceptions of any object type through the creation of new custom objects. A `moldy bread' custom object, for example, normally inherits all the POAGs of `bread' but these can be changed, modified, or removed at the time of object instantiation without affecting other `bread' type objects.

The POAG specification is among the most effective means to collect commonsense knowledge, given that it is performed quite often by the game designer during the creation of scenes.
From a simple POAG definition we may obtain a large amount of knowledge, including interaction semantics between different objects, prerequisites of actions, and the goals commonly associated with such actions. These pieces of commonsense knowledge, are very clearly-structured, and thus easy to assimilate into the knowledge base, due to the fixed framework for defining interaction semantics.

\makefigure{!t}{gecka3dscene2}{Status of an in-game character in the scene. The in-game character has to fulfill the goal of the game, meanwhile he has to avoid the attack of the embedded zombies. }
\subsection{Player Mode}
Besides allowing for the acquisition of knowledge from game designers, GECKA3D enables players of the finished games to be educated in useful ways, all while being entertained.

\makefigure{h}{GECKAxml}{A sample XML output deriving from the creation of a scene in GECKA3D. Actions are collected and encoded according to their semantics.}

In particular, the players use the player mode to control the in-game character to fulfill the predefined goals. The gameplay is also built in 2.5D as similar as the editor mode. As aforementioned, the player has to make decisions quickly to avoid the zombies or monsters. The in-game rooms and corridors are randomly generated to make sure the players have no bias about the spatial layout. Inspired by other games such as Diablo\footnote{https://en.wikipedia.org/wiki/Diablo\_(video\_game)}, only a small portion of view close to in-game character is revealed. The more the character moves, the clearer vision the player achieves.  We also utilize the path finding algorithm for player movement and explorer. When the user clicks to one point in the map, the system automatically finds the shortest path from the user's current position and the target point. The in-game zombies are inserted to put pressure on the player. Note that the zombies chase the player in a turn based manner.

\subsection{Server}
Server is yet another important component of the GECKA3D framework. It is used to store all of the information of the gameplay.
The core functionality include game data uploading from remote clients to server. Note that our editor mode and player mode are deployed in many platforms thanks to the cross-platform characteristic of the chosen game engine. In particular, we use mongoDB and nodejs for server behavior. Http requests are integrated into mobile communication. Data can be transferred from phone to arbitrary server on internet. All the history of game making from the designers and gameplay from the players is stored for further analysis.

\section{Evaluation}
In order to perform a preliminary evaluation of the type and quality of knowledge that GECKA3D can potentially gather, we tested it on 20 university students and staff, who were given the game on tablet and were asked to design a few game scenes over the span of a few hours.Game designers' actions were collected and encoded according to a specific XML format that encodes the semantics associated with such actions (Fig.~\ref{fig:GECKAxml}).
Specific procedures translate these XML files into pieces of common-sense knowledge to be fed to the SenticNet framework \cite{camsen}. Such pieces of commonsense knowledge were manually evaluated (reasonable VS unreasonable) by 5 annotators, resulting in an accuracy of 85.7\%. They also were asked to try the predefined games on the player mode. 18 out of 20 participants highly appreciate the playability and interestingness of the provided player mode which is crucial for attracting enough players to contribute to the embedded human computing. The other two only care about finishing a task as in traditional GWAPs. More tests are due to verify how such new pieces of common-sense knowledge actually improve the reasoning capabilities of the SenticNet framework on specific tasks such as sentiment analysis.

\begin{table}[h]\centering\footnotesize\vspace{5pt}
\begin{tabular}{|x{35pt}|x{28pt}|x{45pt}|x{40pt}|x{28pt}|}\hline
{\bf Item} &{\bf Action} &{\bf Prereq.} &{\bf Outcome} &{\bf Goal}\tabularnewline\hline
blender &blend& orange &orange juice &quench thirst\tabularnewline\hline
bread &cut &knife &bread slices &--\tabularnewline\hline
bread slices &stack &cheese, ham &sandwich &satisfy hunger\tabularnewline\hline
coffee beans &hit &pestle &coffee powder &--\tabularnewline\hline
coffee maker &fill &coffee powder, boiled water &coffee &--\tabularnewline\hline
kettle &fill &water &boiled water &--\tabularnewline\hline
chair &hit &hammer &wood pieces &--\tabularnewline\hline
can &open &can opener &food &satisfy hunger\tabularnewline\hline
towel &cut &scissors &bandage &--\tabularnewline\hline
bag &fill &sand &sandbag &flood control\tabularnewline\hline
\end{tabular}
\caplab{List of most common POAG triples collected during the pilot testing at Singapore Polytechnic.}{tbl:companies}
\end{table}

\section{Conclusion and Future Work}
In this paper, we have introduced the GECKA3D framework which is used for commonsense knowledge acquisition from game designers through the development of serious games. In fact, this not only provides a platform for the acquisition of commonsense knowledge, but also enables the development of games that transfer back the meaningful lessons to gamers. In the future, we aim to update the gameplay, incentives, and interfaces based on analysis of the data resulting from the game, leading to further understanding of the game itself as well as the nexus between game enjoyment and knowledge generation, e.g., finding out what makes providing information fun or what kinds of information are more `fun' than others.

\section{Acknowledgments}
This work was supported by Singapore Ministry of Education under research Grants MOE2012-TIF-2-G-015 and MOE2014-TIF-1-G-007.

\bibliographystyle{./styles/aaai}


\section{Introduction}

Mirror symmetry
started as a mysterious relationship
between complex geometry of a Calabi-Yau 3-fold
and symplectic geometry of another Calabi-Yau 3-fold
called the {\em mirror manifold}.
In 1994, Kontsevich \cite{Kontsevich_HAMS} proposed
a program to understand various mirror phenomena
as a consequence of the following
{\em homological mirror conjecture}:
Calabi-Yau manifolds always come in pairs
in such a way that
the derived category of coherent sheaves
on one manifold is equivalent
as a triangulated category
to the derived Fukaya category of the other.

Although mirror symmetry was first discovered
for Calabi-Yau manifolds,
there are also variants of these phenomena
for other classes of manifolds.
One such example is the Givental's theorem
\cite{Givental_HGMS}
giving integral representations of
$J$-functions
of toric Fano manifolds.
See also \cite{Batyrev_QCRTM}, \cite{Witten_PN2T2D},
\cite{Eguchi-Hori-Xiong} and \cite{Hori-Vafa}.

We can also formulate the homological mirror conjecture
for toric Fano manifolds.
Let $X$ be a toric Fano manifold of dimension $n$.
Then the mirror partner
is
a regular function $W$
on
an algebraic torus $(\ensuremath{\mathbb{C}}^\times)^n$
of the same dimension
equipped with a symplectic structure
(along with an additional data
called a {\em grading}).
Here, the function $W$ is a Newton polynomial
for the convex hull of the generators
of 1-dimensional cone of the fan of $X$.
Coefficients of this polynomial do not matter
as long as they are chosen general enough.
By Kouchnirenko \cite{Kouchnirenko},
$W$ has exactly $\dim H^*(X,\ensuremath{\mathbb{C}})$ critical points.
Take a regular value $t$ of $W$ and
a distinguished basis of
vanishing cycles in $W^{-1}(t)$.
We also have to choose
a {\em grading} and
a {\em spin structure}
on each of these vanishing cycles.
The {\em directed Fukaya category}
$\Fuk^{\rightarrow} W$
of $W$
(along with the choice of gradings
and spin structures)
is an $A_\infty$-category
whose objects are vanishing cycles
and whose morphisms are Floer complexes.
Roughly speaking, the Floer complex
between two vanishing cycles
are the vector space
spanned by intersection points
between them,
and the compositions of morphisms
are given by ``counting polygons.''
By Seidel \cite{Seidel_VC},
the derived category
$D^b \Fuk^{\rightarrow} W$ of
$\Fuk^{\rightarrow} W$
is independent of the choice
of a distinguished basis
of vanishing cycles.

The following is the main result of this paper:

\begin{theorem} \label{th:hms}
The derived category of coherent sheaves
on a toric del Pezzo surface $X$
is equivalent as a triangulated category
to the derived category of
the directed Fukaya category
of the mirror $W$ of $X$;
$$
 D^b \ensuremath{\mathrm{coh}}(X) \cong D^b \Fuk^{\rightarrow} W.
$$
\end{theorem}

Our proof is based on
an explicit computation.
The above Theorem \ref{th:hms}
extends the work of
Seidel \cite{Seidel_VC2}
and
Auroux, Katzarkov and Orlov
\cite{Auroux-Katzarkov-Orlov},
where the cases of
$\ensuremath{\mathbb{P}}^2$ and $\ensuremath{\mathbb{P}}^1 \times \ensuremath{\mathbb{P}}^1$,
and
$\ensuremath{\mathbb{P}}^2$ blown-up at one point
is treated.
See also the paper
by Hori, Iqbal and Vafa
\cite{Hori-Iqbal-Vafa},
where mirror symmetry
for Fano manifolds
are discussed
from a physics point of view.

{\bf Acknowledgements}:
We thank O. Fujino, K. Fukaya,
H. Iritani, T. Kawai,
K. Saito and A. Takahashi
for valuable discussions and comments.
The author is supported by
JSPS Fellowships for Young Scientists
No.15-5561.

\section{Derived category of coherent sheaves}

We describe the structure
of the bounded derived category $\ensuremath{{D^b \mathrm{coh}(Y)}}$
of coherent sheaves
on $Y$ in this section,
where $Y$ is the projective plane
blown-up
at three points $p_1$, $p_2$ and $p_3$
in general position.
Let $\phi : Y \rightarrow \ensuremath{\mathbb{P}}^2$ be this blow-up
and
$E_1$, $E_2$ and $E_3$
be the exceptional divisors
corresponding to
$p_1$, $p_2$ and $p_3$ respectively.
It is a toric surface and
the generators of one-dimensional cones
of its fan is drawn
in Figure \ref{fg:dp3}.

\begin{figure}[H]
\centering
\psfrag{v1}{$v_1$}
\psfrag{v2}{$v_2$}
\psfrag{v3}{$v_3$}
\psfrag{v4}{$v_4$}
\psfrag{v5}{$v_5$}
\psfrag{v6}{$v_6$}
\psfrag{vequalssomething}{$\begin{array}{rcl}
 v_1 & = & (   1   ,   0   ), \\
 v_2 & = & (   0   ,   1   ), \\
 v_3 & = & (   -1  ,  -1   ), \\
 v_4 & = & (   -1  ,   0   ), \\
 v_5 & = & (   0   ,  -1   ), \\
 v_6 & = & (   1   ,   1   ).
	     \end{array}$}
\includegraphics{dp3.eps}
\caption{the toric data of $Y$}
\label{fg:dp3}
\end{figure}

\begin{definition} \label{def:exceptional_collection}
\begin{enumerate}
\item
 An object $\ensuremath{\mathcal{E}}$ in a triangulated category
 is exceptional if
  $$
   \mathop{\mathrm{Ext}}\nolimits^i(\ensuremath{\mathcal{E}},\ensuremath{\mathcal{E}})=\left\{\begin{array}{cl}
      \ensuremath{\mathbb{C}} & \mbox{if $i=0$,}\\
       0  & \mbox{otherwise.}\\
   \end{array}\right.
  $$
\item
 An ordered set of objects $(\ensuremath{\mathcal{E}}_i)_{i=1}^N$
 in a triangulated category
 is an exceptional collection
 if each $\ensuremath{\mathcal{E}}_i$ is exceptional and
 $\mathop{\mathrm{Ext}}\nolimits^k(\ensuremath{\mathcal{E}}_i,\ensuremath{\mathcal{E}}_j)=0$ for any $i>j$
 and for any $k$.
\end{enumerate}

\end{definition}

By combining theorems of Beilinson \cite{Beilinson} and
Orlov \cite{Orlov},
we have the following generators
of $\ensuremath{{D^b \mathrm{coh}(Y)}}$:

\begin{theorem}
Let
\begin{eqnarray*}
\ensuremath{\mathcal{C}} & = &
( \ensuremath{\mathcal{E}}_1, \ensuremath{\mathcal{E}}_2, \ensuremath{\mathcal{E}}_3, \ensuremath{\mathcal{E}}_4, \ensuremath{\mathcal{E}}_5, \ensuremath{\mathcal{E}}_6 )
\end{eqnarray*}
where
\begin{equation} \label{eq:ec}
 \begin{array}{rclrclrcl}
  \ensuremath{\mathcal{E}}_1 & = & \ensuremath{\mathcal{O}}_{E_1}(-1)[-1], &
  \ensuremath{\mathcal{E}}_2 & = & \ensuremath{\mathcal{O}}_{E_2}(-1)[-1], &
  \ensuremath{\mathcal{E}}_3 & = & \ensuremath{\mathcal{O}}_{E_3}(-1)[-1], \\
  \ensuremath{\mathcal{E}}_4 & = & \ensuremath{\phi^* \scO_{\bP^2}(-1)}, &
  \ensuremath{\mathcal{E}}_5 & = & \ensuremath{\phi^* \Omega_{\bP^2}(1)}, &
  \ensuremath{\mathcal{E}}_6 & = & \ensuremath{\scO_Y}. \\
 \end{array}
\end{equation}
Then $\ensuremath{\mathcal{C}}$ is an exceptional collection
generating $\ensuremath{{D^b \mathrm{coh}(Y)}}$.
\end{theorem}
Here, $\ensuremath{\mathcal{O}}_{E_i}(-1)$ is the sheaf supported on
$E_i$ and
is isomorphic to the tautological sheaf $\ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^1}(-1)$
on $E_i \cong \ensuremath{\mathbb{P}}^1$,
$[\bullet]$ is the shift operator in the derived category,
$\phi^*$ is the derived pull-back,
$\ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}(-1)$ is the tautological sheaf
on $\ensuremath{\mathbb{P}}^2$,
and $\Omega_{\ensuremath{\mathbb{P}}^2}(1)$ is the cotangent sheaf of $\ensuremath{\mathbb{P}}^2$
tensored with the hyperplane sheaf
$\ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}(1)$.

\begin{proposition}
All the non-zero $\mathop{\mathrm{Ext}}\nolimits$-groups
within the exceptional collection $\ensuremath{\mathcal{C}}$ are
 \begin{eqnarray*}
  \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_i, \ensuremath{\mathcal{E}}_4)
   & = & \ensuremath{\mathbb{C}} p_i, \\
  \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_i, \ensuremath{\mathcal{E}}_5)
   & = & \mathrm{Ker}(V^\vee \rightarrow (\ensuremath{\mathbb{C}} p_i)^\vee), \\
  \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_i, \ensuremath{\mathcal{E}}_6)
   & = & \ensuremath{\mathbb{C}}, \\
  \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_4, \ensuremath{\mathcal{E}}_5)
   & = & \Lambda^2 V^\vee, \\
  \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_4, \ensuremath{\mathcal{E}}_6)
   & = & V^\vee, \\
  \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_5, \ensuremath{\mathcal{E}}_6)
   & = & V,
 \end{eqnarray*}
where $i=1,2,3$,
$V \cong \ensuremath{\mathbb{C}}^3$ is the three-dimensional vector space such that
$\ensuremath{\mathbb{P}}^2 = \ensuremath{\mathbb{P}}(V)$,
the check denotes the dual vector space,
$\ensuremath{\mathbb{C}} p_i \subset V$ is the one-dimensional subspace
corresponding to $p_i \in \ensuremath{\mathbb{P}}(V)$,
and the map $V^\vee \rightarrow (\ensuremath{\mathbb{C}} p_i)^\vee$
is the dual of the inclusion
$\ensuremath{\mathbb{C}} p_i \hookrightarrow V$.
Compositions of morphisms
are given by
$$
\begin{array}{ccc}
 \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_i, \ensuremath{\scE_4}) \times \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_4}, \ensuremath{\scE_5})
   & \longrightarrow & \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_i, \ensuremath{\scE_5})\\
 \rotatebox{90}{$\in$} & & \rotatebox{90}{$\in$} \\
 (v, \omega ) & \longmapsto & -\iota_v \omega
\end{array},
$$
$$
\begin{array}{ccc}
 \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_i, \ensuremath{\scE_4}) \times \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_4}, \ensuremath{\scE_6})
   & \longrightarrow & \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_i, \ensuremath{\scE_6})\\
 \rotatebox{90}{$\in$} & & \rotatebox{90}{$\in$} \\
 (v, \omega ) & \longmapsto & \omega(v)
\end{array},
$$
$$
\begin{array}{ccc}
 \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_i, \ensuremath{\scE_5}) \times \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_5}, \ensuremath{\scE_6})
   & \longrightarrow & \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_i, \ensuremath{\scE_6})\\
 \rotatebox{90}{$\in$} & & \rotatebox{90}{$\in$} \\
 (\omega, v ) & \longmapsto & \omega(v)
\end{array},
$$
$$
\begin{array}{ccc}
 \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_4}, \ensuremath{\scE_5}) \times \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_5}, \ensuremath{\scE_6})
   & \longrightarrow & \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_4}, \ensuremath{\scE_6})\\
 \rotatebox{90}{$\in$} & & \rotatebox{90}{$\in$} \\
 (\omega, v ) & \longmapsto & \iota_v \omega
\end{array}
$$
where $\iota_v : \wedge^2 V^\vee \rightarrow V^\vee$
for $v \in V$ is the interior product.
\end{proposition}

\proof
All the computations reduce to those on $\ensuremath{\mathbb{P}}^2$
in the following way:
Let $\ensuremath{\mathcal{E}}$ and $\ensuremath{\mathcal{F}}$ be vector bundles
on $\ensuremath{\mathbb{P}}^2$. Then
\begin{eqnarray*}
 \mathop{\ensuremath{\mathbb{R}}\mathrm{Hom}}\nolimits(\phi^* \ensuremath{\mathcal{E}}, \phi^* \ensuremath{\mathcal{F}})
 & = & \mathop{\ensuremath{\mathbb{R}}\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}, \bR \phi_* \phi^* \ensuremath{\mathcal{F}}) \\
 & = & \mathop{\ensuremath{\mathbb{R}}\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}, \ensuremath{\mathcal{F}} \otimes \bR \phi_* \ensuremath{\mathcal{O}}_Y) \\
 & = & \mathop{\ensuremath{\mathbb{R}}\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}, \ensuremath{\mathcal{F}}),
\end{eqnarray*}
\begin{eqnarray*}
 \mathop{\ensuremath{\mathbb{R}}\mathrm{Hom}}\nolimits(\phi^* \ensuremath{\mathcal{E}}, \ensuremath{\mathcal{O}}_{E_i}(-1))
 & = & \mathop{\ensuremath{\mathbb{R}}\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}, \bR \phi_* \ensuremath{\mathcal{O}}_{E_i}(-1)) \\
 & = & 0,
\end{eqnarray*}
and
\begin{eqnarray*}
\mathop{\ensuremath{\mathbb{R}}\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{O}}_{E_i}(-1)[-1], \phi^* \ensuremath{\mathcal{E}})
 & = & \mathop{\ensuremath{\mathbb{R}}\mathrm{Hom}}\nolimits( \{ \ensuremath{\mathcal{O}}_Y \rightarrow \ensuremath{\mathcal{O}}_Y(E_i) \},
   \phi^* \ensuremath{\mathcal{E}} ) \\
 & = & \mathop{\ensuremath{\mathbb{R}}\Gamma}\nolimits( \{ \ensuremath{\mathcal{O}}_Y(-E_i) \rightarrow \ensuremath{\mathcal{O}}_Y \} \otimes
   \phi^* \ensuremath{\mathcal{E}} ) \\
 & = & \mathop{\ensuremath{\mathbb{R}}\Gamma}\nolimits( \{ \bR \phi_* \ensuremath{\mathcal{O}}_Y(-E_i) \rightarrow
   \bR \phi_* \ensuremath{\mathcal{O}}_Y \} \otimes \ensuremath{\mathcal{E}} ) \\
 & = & \mathop{\ensuremath{\mathbb{R}}\Gamma}\nolimits( \{ \ensuremath{\mathcal{I}}_{p_i} \rightarrow \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2} \}
   \otimes \ensuremath{\mathcal{E}} ) \\
 & = & \mathop{\ensuremath{\mathbb{R}}\Gamma}\nolimits( \ensuremath{\mathcal{O}}_{p_i} \otimes \ensuremath{\mathcal{E}} ) \\
 & = & \ensuremath{\mathcal{E}}|_{p_i},
\end{eqnarray*}
where
$\ensuremath{\mathcal{I}}_{p_i}$ is the ideal sheaf of $p_i$
and the last line is the fiber at $p_i$.
Here, we have used the exact sequence
$$
0 \rightarrow \ensuremath{\mathcal{O}}_Y \rightarrow \ensuremath{\mathcal{O}}_Y(E_i)
  \rightarrow \ensuremath{\mathcal{O}}_{E_i}(-1) \rightarrow 0.
$$
We can further use the exact sequence
$$
0 \rightarrow \Omega_{\ensuremath{\mathbb{P}}^2}(1) \rightarrow V^\vee \otimes \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}
 \rightarrow \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}(1) \rightarrow 0
$$
to reduce the computations involving $\Omega_{\ensuremath{\mathbb{P}}^2}(1)$
to those involving $\ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}(i)$, $i \in \ensuremath{\mathbb{Z}}$:
\begin{eqnarray*}
\mathop{\ensuremath{\mathbb{R}}\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{O}}_{E_i}(-1)[-1], \ensuremath{\phi^* \Omega_{\bP^2}(1)})
 & = & \mathop{\ensuremath{\mathbb{R}}\Gamma}\nolimits( \ensuremath{\mathcal{O}}_{p_i} \otimes
         \{V^\vee \otimes \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}
                \rightarrow \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}(1) \} ) \\
 & = & \{V^\vee \rightarrow (\ensuremath{\mathbb{C}} p_i)^\vee \},
\end{eqnarray*}
\begin{eqnarray*}
\mathop{\ensuremath{\mathbb{R}}\mathrm{Hom}}\nolimits(\ensuremath{\phi^* \scO_{\bP^2}(-1)}, \ensuremath{\phi^* \Omega_{\bP^2}(1)})
 & = & \mathop{\ensuremath{\mathbb{R}}\mathrm{Hom}}\nolimits( \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}(-1),
         \{ V^\vee \otimes \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}
                 \rightarrow \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}(1) \} ) \\
 & = & \mathop{\ensuremath{\mathbb{R}}\Gamma}\nolimits( \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}(1) \otimes
         \{ V^\vee \otimes \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}
                 \rightarrow \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}(1) \} ) \\
 & = & \mathop{\ensuremath{\mathbb{R}}\Gamma}\nolimits( \{ V^\vee \otimes \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}(1)
                 \rightarrow \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}(2) \} ) \\
 & = & \{V^\vee \otimes V^\vee \rightarrow \ensuremath{\mathop{\mathrm Sym}\nolimits}^2 V^\vee \} \\
 & = & \wedge^2 V^\vee,
\end{eqnarray*}
\begin{eqnarray*}
\mathop{\ensuremath{\mathbb{R}}\mathrm{Hom}}\nolimits(\ensuremath{\phi^* \Omega_{\bP^2}(1)}, \ensuremath{\scO_Y})
 & = & \mathop{\ensuremath{\mathbb{R}}\mathrm{Hom}}\nolimits( \{ V^\vee \otimes \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}
                 \rightarrow \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}(1) \},
               \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2} ) \\
 & = & \mathop{\ensuremath{\mathbb{R}}\Gamma}\nolimits( \{ \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2}(-1) \rightarrow
                  V \otimes \ensuremath{\mathcal{O}}_{\ensuremath{\mathbb{P}}^2} \} ) \\
 & = & V.
\end{eqnarray*}
Compositions of morphisms can be
easily read off
from the above computations.
\qed

\section{Fukaya category} \label{Fuk}

First we recall the definition of an $A_\infty$-category.
For a $\ensuremath{\mathbb{Z}}$-graded vector space $V$
and $i \in \ensuremath{\mathbb{Z}}$,
$V[i]$ denotes the shift of $V$ by $i$;
$V[i]^j=V^{i+j}.$

\begin{definition} \label{def:a_infinity}
An $A_\infty$-category $\ensuremath{\mathcal{A}}$ consists of
\begin{enumerate}
\item
the set of objects $\mathfrak{Ob}(\ensuremath{\mathcal{A}})$,
\item
for any $c_1,\; c_2 \in \mathfrak{Ob}(\ensuremath{\mathcal{A}})$,
a $\ensuremath{\mathbb{Z}}$-graded $\ensuremath{\mathbb{C}}$-vector space $\hom_{\ensuremath{\mathcal{A}}}(c_1, c_2)$
called
the set of morphisms,
\item
for any positive integer $k$
and for any set of objects $\{c_i\}_{i=0}^k$,
the composition
$$\
 \mathfrak{m}_k : \hom_{\ensuremath{\mathcal{A}}}(c_0,c_1)[1]\otimes \cdots
                \otimes \hom_{\ensuremath{\mathcal{A}}}(c_{k-1},c_k)[1]
                   \longrightarrow \hom_{\ensuremath{\mathcal{A}}}(c_0,c_k)[1]
$$
which is a linear map
of degree $1$
\end{enumerate}
such that
for any positive integer $k$,
any set of objects $\{c_i\}_{i=0}^k$
and any set of morphisms $\{a_i\}_{i=1}^k$,
$a_i \in \hom_{\ensuremath{\mathcal{A}}}(c_{i-1},c_i)$,
the following {\em $A_\infty$-relations} holds:
\begin{eqnarray*} \label{eq:a_infinity}
 \sum_{i=0}^{k-1} \sum_{j=i+1}^k
  (-1)^{\mathrm{deg} a_1 + \cdots + \mathrm{deg} a_i}
  \mathfrak{m}_{i-j+k+1}(a_1 \otimes \cdots \otimes a_i \\
   \otimes
    \mathfrak{m}_{j-i}(a_{i+1} \otimes \cdots \otimes a_j)
   \otimes
    a_{j+1} \otimes \cdots \otimes a_k ) = 0.
\end{eqnarray*}
Here degrees are counted after shifts, i.e., if $a \in V^i$,
$\mathrm{deg} a = i-1$ in $V[1]$.
Since $\mathfrak{m}_1^2=0$, we define
$$
 \mathop{\mathrm{Hom}}\nolimits_{\ensuremath{\mathcal{A}}}(c_1,c_2) = H^0(\hom_{\ensuremath{\mathcal{A}}}(c_1,c_2),\mathfrak{m}_1).
$$
\end{definition}

The Fukaya category
of Lagrangian submanifolds
in a symplectic manifold
is defined in
\cite{Fukaya-Oh-Ohta-Ono}.
We use the following adaptation
for exact Morse fibrations
by Seidel \cite{Seidel_VC}.
Let $W: Z \rightarrow \ensuremath{\mathbb{C}}$
be a regular function
on an affine algebraic manifold $Z$
of complex dimension $n$
with a K\"{a}hler structure.
Assume the following conditions:
\begin{itemize}
 \item The K\"{a}hler metric is complete.
 \item The symplectic form $\omega$
  of $Z$ is exact,
  i.e., there exists a one form $\theta$ on $Z$
  such that $\omega = d \theta$.
 \item At any critical points of $W$,
  the Hessian of $W$ is non-degenerate.
 \item All the critical values are distinct.
\end{itemize}
Such $W$ gives rise to an exact Morse fibration
in the terminology of \cite{Seidel_VC}.
Using the K\"{a}hler structure,
we can define the lift
$\widetilde{c}_p : [0,1] \rightarrow Z$
of a path
$c : [0,1] \rightarrow \ensuremath{\mathbb{C}}$
starting from a point $p \in Z$
such that $W(p) = c(0)$
by using the horizontal distribution
defined as the orthogonal complement
of the tangent space
along the fiber of $W$.

Assume that the origin is
a regular value of $W$ and
fix an order $(p_i)_{i=1}^N$
on the set of critical points of $W$.
A distinguished set $(c_i)_{i=1}^N$
of vanishing paths is a set of smooth paths
$c_i : [0,1] \rightarrow \ensuremath{\mathbb{C}}$
satisfying
 \begin{enumerate}
  \item $c_i(0) = 0$, $c_i(1) = W(p_i)$,
  \item $c_i$ has no self-intersection,
  \item images of $c_i$ and $c_j$ intersects only at the origin,
  \item $c_i'(0) \neq 0$, and
  \item $\arg c_{i+1}'(0) < \arg c_i'(0)$, $i=1,\ldots N-1$,
         for some choice of the branch of $\arg(\bullet)$.
 \end{enumerate}
Given a distinguished set
$(c_i)_{i=1}^N$ of vanishing paths,
the corresponding vanishing cycles
$(C_i)_{i=1}^N$
are defined by
$$
 C_i = \{p \in W^{-1}(0) \; | \;
           \lim_{t \rightarrow 1} \widetilde{c}_p(t) = p_i\}.
$$
They are Lagrangian submanifolds
of $W^{-1}(0)$.

The {\em directed Fukaya category} $\Fuk^{\rightarrow} W$
for $W$ is roughly an $A_\infty$-category
whose objects are vanishing cycles
and whose morphisms are Lagrangian intersection
Floer complexes.
To define a $\ensuremath{\mathbb{Z}}$-grading
on the Floer complex,
we need the concept of {\em grading}
on Lagrangian submanifolds
introduced by Kontsevich \cite{Kontsevich_HAMS},
which we now recall.
See also Seidel \cite{S}.

Let $(M,\omega)$ be a symplectic manifold
of dimension $2n$.
An almost complex structure on $M$
is a section $J \in \Gamma(M, \mathop{\mathrm{End}}\nolimits(T M))$
such that $J^2 = -\ensuremath{\mathop{\mathrm{id}}}$.
$J$ is called $\omega$-compatible
if $g(V_1,V_2) = \omega(V_1, J V_2)$
defines a Hermitian metric on the tangent bundle.
Fix an $\omega$-compatible almost complex structure $J$
of $M$ and
let $\ensuremath{\mathcal{S}}$ be the principal $U(1)$-bundle
associated to the complex line bundle
$(\Lambda^n (T^*M, J))^{\otimes 2}$.
The fiber of $\ensuremath{\mathcal{S}}$ at $p \in M$ is
$(\Lambda^n (T_p^* M, J))^{\otimes 2} / \ensuremath{\mathbb{R}}^{> 0}$.
We assume that the first Chern class
of this complex line bundle vanishes,
so that $\ensuremath{\mathcal{S}}$ has a section.
A grading of $M$ is
a choice of a section $\Theta : M \rightarrow \ensuremath{\mathcal{S}}$.
Fix a grading $\Theta$ on $M$.
Let $\mathcal{L}ag_M \rightarrow M$ be
the Lagrangian Grassmannian bundle on $M$,
whose fiber at $p \in M$
is the Grassmannian of Lagrangian subspaces
in the symplectic vector space $T_p M$.
Define $\det_\Theta^2 : \mathcal{L}ag_M \rightarrow U(1)$ as follows:
For a Lagrangian subspace $L \subset T_p M$,
pick any basis $\{e_i\}_{i=1}^n$ of $L$
and take the square of their exterior product;
$(e_1 \wedge e_2 \wedge \cdots \wedge e_n)^{\otimes 2}
\in (\Lambda^n (T_p M, J))^{\otimes 2}$.
$\det_\Theta^2(L)$ is the image of this element by $\Theta(p)$.
A Lagrangian submanifold $L \subset M$
gives a canonical section
$s_L : L \ni p \mapsto T_p L \in \mathcal{L}ag_M|_p$.
Denote the composition of $s_L$ and $\det_\Theta^2$ by $\phi_L$.
A grading of a Lagrangian submanifold
is a lift $\widetilde{\phi_L} : L \rightarrow \ensuremath{\mathbb{R}}$
of $\phi_L$ to the universal cover $\ensuremath{\mathbb{R}}$ of
$U(1) \cong \ensuremath{\mathbb{R}}/\ensuremath{\mathbb{Z}}$.

Now we define the Maslov index.
A smooth path
$\Lambda : [0,1] \rightarrow \mathcal{L}ag(V,\omega)$
in the Lagrangian Grassmannian
of a fixed symplectic vector space $(V, \omega)$
of dimension $2 n$
is called {\em crossingless}
if $\Lambda(0) \cap \Lambda(t) = \Lambda(0) \cap \Lambda(1)$
for all $t \in (0,1]$.
For a crossingless path $\Lambda_t$,
its differential $\Lambda'(0)$ at $t=0$
gives an element of the tangent space
$T_{\Lambda(0)} \mathcal{L}ag(V,\omega) \subset T_{\Lambda(0)} \ensuremath{\mathrm{Gr}}(n,V)
  \cong \mathop{\mathrm{Hom}}\nolimits(\Lambda(0),V/\Lambda(0))$
where $\ensuremath{\mathrm{Gr}}(n,V)$ is the Grassmannian of $n$-dimensional
subspaces in $V$.
The composition of $\Lambda'(0)$ and $\omega$
defines a quadratic form
$\Lambda(0) \ni v \mapsto \omega(v, \Lambda'(0) v)$
on $\Lambda(0)$,
which descends to a quadratic form on
$\Lambda(0)/\Lambda(0) \cap \Lambda(1)$
since it vanishes on $\Lambda(0) \cap \Lambda(1)$.
The resulting form on $\Lambda(0)/\Lambda(0) \cap \Lambda(1)$
is called the {\em crossing form} \cite{Robbin-Salamon}.
For an intersection $p \in L_1 \cup L_2$
of two graded Lagrangian submanifolds
$(L_1, \widetilde{\phi_{L_1}})$ and $(L_2, \widetilde{\phi_{L_2}})$,
its Maslov index is defined as follows:
Choose a crossingless path
$\Lambda : [0,1] \rightarrow \mathcal{L}ag_M|_p$
from $T_p L_0$ to $T_p L_1$
such that the corresponding crossing form
at $t=0$ is negative definite.
There is a unique lift
$\widetilde{\alpha} : [0,1] \rightarrow \ensuremath{\mathbb{R}}$
of the composition $\alpha$ of
$\Lambda : [0,1] \rightarrow \mathcal{L}ag_M|_p$ and
$\det_\Theta^2(p) : \mathcal{L}ag_M|_p \rightarrow U(1)$
such that $\widetilde{\alpha}(0)=\widetilde{\phi_{L_0}}(p)$,
and the Maslov index $I(p)$ of the intersection
$p$ of two graded Lagrangian submanifolds
is defined by
$$
 I(p) = \widetilde{\phi_{L_1}}(p) - \widetilde{\alpha}(1).
$$

Now we come back to our exact Morse fibration $W$.
A relative Maslov map
is a section $\Theta$
of the second tensor power
$(\Omega_{Z / \ensuremath{\mathbb{C}}}^{n-1})^{\otimes 2}$
of the top exterior product
of the relative cotangent bundle
away from the critical points.
Since $0$ is a regular value,
the restriction of $\Theta$
to $W^{-1}(0)$ gives a grading
of $W^{-1}(0)$.
Assume that
all the vanishing cycles can be graded,
and fix a grading on each vanishing cycle.
We also need to choose a spin structure
on each vanishing cycles
in order to orient
the moduli spaces of
pseudoholomorphic maps
\cite{Fukaya-Oh-Ohta-Ono}.
Since
each vanishing cycle is homeomorphic
to a sphere,
the choice of a spin structure
is unique
except for $\dim_\ensuremath{\mathbb{C}} W^{-1}(0) = 1$,
where one has as many as
$H^1(S^1,\ensuremath{\mathbb{Z}}/2 \ensuremath{\mathbb{Z}})=\ensuremath{\mathbb{Z}} / 2 \ensuremath{\mathbb{Z}}$
choices.
We take the non-trivial spin structure
in such a case.
Let $C_i^\flat$ denote
the vanishing cycle $C_i$
endowed with the above grading and spin structure.
We assume that vanishing cycles intersects each other
transversally.
This condition can always be met
by moving vanishing cycles
within their Hamiltonian isotopy classes
if necessary.

\begin{definition}
Given a function $W$ on an affine K\"{a}hler manifold $Z$
together with a relative Maslov map $\Theta$
and a choice of a distinguished basis
of vanishing cycles
$(C_i^\flat)_{i=1}^N$
with gradings and spin structures,
its directed Fukaya category $\Fuk^{\rightarrow} W$
is an $A_\infty$-category such that
\begin{itemize}
 \item the set of objects
        is the distinguished basis of vanishing cycles;
        $$ \mathfrak{Ob}(\Fuk^{\rightarrow} W) = (C_1^\flat, \ldots, C_N^\flat),$$
 \item the set of morphisms between
        $C_i^\flat$ and $C_j^\flat$ is
        the $\ensuremath{\mathbb{Z}}$-graded vector space
$$
{\hom}_{\FukW}(C_i^\flat, C_j^\flat)
 = \left\{
  \begin{array}{lc}
   0 & i>j, \\
   \ensuremath{\mathbb{C}} \cdot \ensuremath{\mathop{\mathrm{id}}}_{C_i^\flat} & i=j, \\
   \bigoplus_{p \in C_i \cap C_j} \ensuremath{\mathop{\mathrm{span}}}_{\ensuremath{\mathbb{C}}}\{p\} & i<j
  \end{array}
  \right.
$$
      where $\mathrm{deg} p= I(p)$ (the Maslov index), and
 \item for a positive integer $k$,
        a set of objects
        $(C_{i_0}^\flat,\ldots,C_{i_k}^\flat)$
        and morphisms $p_l \in C_{i_{l-1}} \cap C_{i_l}$
        for $l = 1,\ldots,k$,
        the composition $\mathfrak{m}_k$ is given by
$$
 \mathfrak{m}_k(p_1,\ldots,p_k) = \sum_{p_0 \in C_{i_0} \cap C_{i_k}}
     \# \overline{\ensuremath{\mathcal{M}}}_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k) p_0.
$$
\end{itemize}
\end{definition}
Here, $\overline{\ensuremath{\mathcal{M}}}_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k)$
is the stable compactification of \\
$\ensuremath{\mathcal{M}}_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k)$
defined as follows:
A disk with $k+1$ marked points
on the boundary
is a pair $(D^2,(z_0,\ldots,z_k))$
of a closed unit disk
$D^2=\{z \in \ensuremath{\mathbb{C}} \; | \; |z| \leq 1\}$
and an ordered set
$(z_0,\ldots,z_k)$
of $k+1$ points on the boundary
respecting the cyclic order.
Let $\partial_l D^2 \in \partial D^2$
be the interval
between $z_l$ and $z_{l+1}$,
where we set $z_{k+1} = z_0$.
Fix an almost complex structure $J$ on $W^{-1}(0)$.
A smooth map $\varphi : D^2 \rightarrow M$
is called pseudoholomorphic if
$$
d \varphi \circ J_{D^2} = J \circ d \varphi
$$
where $J_{D^2}$ is the canonical complex structure
on $D^2$.
$\ensuremath{\mathcal{M}}_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k)$ is
the moduli space of pairs
$((D^2,(z_0,\ldots,z_k)),\varphi)$
such that
\begin{enumerate}
 \item $(D^2,(z_0,\ldots,z_k))$ is a disk
   with $k+1$ marked points on the boundary,
 \item $\varphi : D^2 \rightarrow W^{-1}(0)$
   is a pseudoholomorphic map,
 \item $\varphi(\partial_l D^2) \subset C_{i_l}$ for $l=0,\ldots,k$, and
 \item $\varphi(z_l) = p_l$ for $l=0,\ldots,k$.
\end{enumerate}

Although $\overline{\ensuremath{\mathcal{M}}}_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k)$
is not an honest manifold in general
due to subtleties in the definition of
the moduli space of pseudoholomorphic maps,
it has a Kuranishi structure with corners
and it makes sense to ``count'' numbers
$\# \overline{\ensuremath{\mathcal{M}}}_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k)$
of points.
See \cite{Fukaya-Oh-Ohta-Ono}
for details.
These numbers are counted with signs
determined by the orientation of the moduli space,
which in turn is determined
by the spin structures on vanishing cycles.
$\# \overline{\ensuremath{\mathcal{M}}}_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k)$
is zero if \\
$\dim \overline{\ensuremath{\mathcal{M}}}_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k) \neq 0$.

These moduli spaces must be compact
so that it makes sense
to count the numbers of points.
To prove compactness of the moduli space,
one needs the boundedness
of the energy.
Usually, this is achieved
by introducing
the Novikov ring
as the coefficient ring
of the Floer complex
in order to control energies
of pseudoholomorphic maps.
This is not necessary in our case
since vanishing cycles are exact Lagrangian submanifolds,
i.e., $[\theta] = 0 \in H^1(C_i,\ensuremath{\mathbb{R}})$
for the primitive $\theta$ of $\omega$;
$\omega = d \theta$.
To see this,
let $C$ be the vanishing cycle
corresponding to a vanishing path
$c : [0,1] \rightarrow \ensuremath{\mathbb{C}}$.
Then $C$ bounds the disk
$D = \bigcup_{p \in C} \widetilde{c_p}([0,1])$;
$\partial D = C$.
Since
$d \theta|_D = \omega|_D = 0$,
$[\theta] = 0 \in H^1(\partial D,\ensuremath{\mathbb{R}})$.
Therefore,
there exists a function $K_i$
on each vanishing cycles
such that
$\theta|_{C_i} = d K_i$.
Then for any $\varphi \in
\overline{\ensuremath{\mathcal{M}}}_{k+1}(C_{i_0},\ldots,C_{i_k};p_0,\ldots,p_k)$,
the energy of $\varphi$ is
\begin{eqnarray*}
 E(\varphi) & = & \int_{D^2} \varphi^*\omega
  =  \int_{\partial D^2} \varphi^* \theta
  =  \sum_{l=0}^k \int_{\partial_i D^2} \varphi^*\theta \\
 &=& \sum_{l=0}^k (K_{i_l}(p_{l+1})- K_{i_l}(p_l))
\end{eqnarray*}
and does not depend on the homotopy class of $\varphi$.
This also implies the vanishing of $\mathfrak{m}_0$.
Non-compactness of $W^{-1}(0)$
does not cause any problem either.
Since $W^{-1}(0)$ is Stein,
the maximum principle prevents
pseudoholomorphic  disks
from running away to infinity,
assuring the compactness of the moduli space.

Seidel proved that although
the directed Fukaya category $\Fuk^{\rightarrow} W$
depends on the choice of the distinguished basis
of vanishing cycles,
different choices are related by mutations,
hence its derived category
is invariant.

Now we explain the mirror construction
of toric Fano manifolds
after Givental \cite{Givental_HGMS}.
Given a fan of an $n$-dimensional toric Fano manifold,
let $\{v_i\}_{i=1}^r$,
$v_i = (v_{i1},\ldots,v_{in}) \in \ensuremath{\mathbb{Z}}^n$,
be the set of generators of
its one-dimensional cones.
Then the mirror object
for this
toric Fano manifold is
the regular function
\begin{equation} \label{eq:generalW}
 W(x_1,\ldots,x_n)=\sum_{i=1}^r q_i x_1^{v_{i1}} \cdots x_n^{v_{in}}
\end{equation}
on the algebraic torus $\mathrm{Spec} \ensuremath{\mathbb{C}}[x_i^{\pm 1}]_{i=1}^n$.
Here, $q_i$'s are parameters
corresponding to the deformation
of symplectic structures on the toric Fano manifold.

Therefore, the mirror of our $Y$
is the regular function
\begin{equation} \label{eq:W}
 W(x,y) = q_1 x + q_2 y + \frac{q_3}{x y}
          + \frac{q_4}{x} + \frac{q_5}{y} + q_6 x y
\end{equation}
on the algebraic torus
$(\ensuremath{\mathbb{C}}^\times)^2 = \mathrm{Spec} \ensuremath{\mathbb{C}}[x,x^{-1},y,y^{-1}]$
(See Figure \ref{fg:dp3}).
The Fukaya category does not depend
on a general choice of $q_i$'s.
We have used
$(q_1,q_2,q_3,q_4,q_5,q_6) = (1,1,1,0.215,0.25,0.3)$
to draw the figures appearing below.
Equip $(\ensuremath{\mathbb{C}}^\times)^2$
with the symplectic form
$\frac{d|x|}{|x|} \wedge d(\arg x) + \frac{d|y|}{|y|} \wedge d(\arg y)$
and
the relative Maslov map
$$
 \Theta = (\ensuremath{\mathop{\mathrm{Res}}} \frac{dx \wedge dy}{x y W(x,y)})^{\otimes 2}
        = (\frac{dx}{\partial_y (x y W(x,y))})^{\otimes 2}.
$$
$W^{-1}(0)$ is an affine elliptic curve,
which can be compactified by adding six points.
We depict the critical values of $W$
and our choice of a distinguished set
of vanishing paths
in Figure \ref{fg:vanishing_paths}.
Figure \ref{fg:vanishing_cycles}
shows the corresponding
vanishing cycles.

\begin{figure}[htbp]
\centering
\psfrag{c1}{$c_1$}
\psfrag{c2}{$c_2$}
\psfrag{c3}{$c_3$}
\psfrag{c4}{$c_4$}
\psfrag{c5}{$c_5$}
\psfrag{c6}{$c_6$}
\includegraphics{vanishing_paths.eps}
\caption{Distinguished basis of vanishing paths}
\label{fg:vanishing_paths}
\end{figure}

\begin{figure}[htbp]
\centering
\psfrag{C3}{$C_1$}
\psfrag{C4}{$C_2$}
\psfrag{C5}{$C_3$}
\psfrag{C6}{$C_4$}
\psfrag{C7}{$C_5$}
\psfrag{C8}{$C_6$}
\psfrag{x1}{$x_1$}
\psfrag{x2}{$x_2$}
\psfrag{x3}{$x_3$}
\psfrag{x12*}{$x_1^\vee \wedge x_2^\vee$}
\psfrag{x23*}{$x_2^\vee \wedge x_3^\vee$}
\psfrag{x31*}{$x_3^\vee \wedge x_1^\vee$}
\psfrag{x1*}{$x_1^\vee$}
\psfrag{-x1*}{$-x_1^\vee$}
\psfrag{x2*}{$x_2^\vee$}
\psfrag{x3*}{$x_3^\vee$}
\psfrag{e1}{$e_{1}$}
\psfrag{e2}{$e_{2}$}
\psfrag{e3}{$e_{3}$}
\scalebox{1}{\includegraphics{dp3_vanishing_cycles.eps}}
\caption{Vanishing cycles in the fiber at the origin}
\label{fg:vanishing_cycles}
\end{figure}

The opposite sides of the square
in Figure \ref{fg:vanishing_cycles}
are identified
to form a two-dimensional torus.
The open circles denote the points
which are missing
due to the non-compactness
of $(\ensuremath{\mathbb{C}}^\times)^2$.
We have drawn Figure \ref{fg:vanishing_cycles}
in the following way:
First regard $W^{-1}(t)$ as a branched double
cover of $\ensuremath{\mathbb{C}}^\times$
by the projection
$\pi_t : W^{-1}(t) \ni (x,y) \mapsto y \in \ensuremath{\mathbb{C}}^\times$.
For any $i =1,\ldots,6$,
the branch points of $\pi_{c_i(t)}$ moves in $\ensuremath{\mathbb{C}}^\times$
as we vary $t$,
until two of them finally collides
at $t=1$.
The vanishing cycle $C_i$ is
the circle in $W^{-1}(0)$
over this trajectory of collision.
This determines the vanishing cycle $C_i$
up to isotopy.
These vanishing cycles
can be straightened within their Hamiltonian isotopy classes.
Here, being straight refers to
the flat metric on the torus
$\overline{W^{-1}(0)}$,
where $\overline{\bullet}$ denotes
the completion.
Note that this flat metric on
$\overline{W^{-1}(0)}$
has nothing to do
with the K\"{a}hler metric
on $W^{-1}(0)$
induced from that of
$(\ensuremath{\mathbb{C}}^\times)^2$.
This determines $C_i$ up to translation.
This translational ambiguity can be fixed
by imposing exactness,
which requires the knowledge of the primitive
$\theta$ of the symplectic form $\omega$
on $W^{-1}(0)$.
We do not try to carry this out
since translations of straight $C_i$'s
do not alter the combinatorial structure
of intersections,
which is all we need in our computation of the
Fukaya category below.
The grading on $W^{-1}(0)$
given by $\Theta$
is the grading
coming from the restriction
of the second tensor power
of the holomorphic 1-form
on $\overline{W^{-1}(0)}$.
Then
one can choose a grading $\widetilde{\phi_{C_i}}$
on each vanishing cycle $C_i$
so that
all the Maslov indices
of $C_i \cap C_j$
is zero for $i < j$.
To give a spin structure on $C_i$
is the same as to give a two-fold covering of $C_i$.
We take the non-trivial cover
such that two branches interchange
on the black dots in Figure \ref{fg:vanishing_cycles}.
Different choices for spin structures
lead to different categories,
and it turns out that
the above choice gives a category
derived equivalent to
the category of coherent sheaves
on the del Pezzo surface $Y$.
Let $C_i^\flat$ denote the vanishing cycle $C_i$
with the above grading and spin structure.
Since the Maslov index
of all the intersection points are zero,
the sign for the counting of a triangle
is $-1$ if the boundary of the triangle
hits odd numbers of these dots,
and $+1$ otherwise
(see Seidel \cite{Seidel_K3}).
Note that although the choice of spin structures
is important,
the choice of the positions
of the black dots is irrelevant.
The change of the signs
caused by a change of the positions
of the black dots
can be absorbed
by a redefinition of
the signs of the basis for the Floer cohomologies.
$\mathfrak{m}_k$ is non-zero only for $k=2$
(polygons with more than three edges
in Figure \ref{fg:vanishing_cycles}
do not contribute
since ${\hom}_{\FukW}(C^\flat_i,C^\flat_j) = 0$ for $i > j$).

\begin{theorem} \label{th:commutativity}
There exists an isomorphism
$$
 \phi_{ij} : \mathop{\mathrm{Hom}}\nolimits_\ensuremath{{D^b \mathrm{coh}(Y)}}(\ensuremath{\mathcal{E}}_i, \ensuremath{\mathcal{E}}_j)
       \rightarrow \Hom_{\FukW}(C_i^\flat,C_j^\flat)
$$
as a $\ensuremath{\mathbb{C}}$-vector space
for $i,j=1,\ldots,6$
such that
the diagrams
\begin{equation} \label{eq:diagram}
\begin{CD}
 \mathop{\mathrm{Hom}}\nolimits_\ensuremath{{D^b \mathrm{coh}(Y)}}(\ensuremath{\mathcal{E}}_i, \ensuremath{\mathcal{E}}_j) \times \mathop{\mathrm{Hom}}\nolimits_\ensuremath{{D^b \mathrm{coh}(Y)}}(\ensuremath{\mathcal{E}}_j, \ensuremath{\mathcal{E}}_k)
   @>>> \mathop{\mathrm{Hom}}\nolimits_\ensuremath{{D^b \mathrm{coh}(Y)}}(\ensuremath{\mathcal{E}}_i, \ensuremath{\mathcal{E}}_k) \\
 @VV{\phi_{ij} \times \phi_{jk}}V @VV{\phi_{ik}}V \\
 \Hom_{\FukW}(C_i^\flat,C_j^\flat) \times \Hom_{\FukW}(C_j^\flat,C_k^\flat)
   @>>> \Hom_{\FukW}(C_i^\flat,C_k^\flat)
\end{CD}
\end{equation}
commute.
\end{theorem}
\proof
We omit the lower suffix of $\mathop{\mathrm{Hom}}\nolimits(\bullet,\bullet)$
since there is no danger of confusion.
We construct the above $\phi_{ij}$'s explicitly.
Since $\mathfrak{m}_1=0$,
$\mathop{\mathrm{Hom}}\nolimits(C_i^\flat,C_j^\flat)$
is isomorphic to $\hom(C_i^\flat,C_j^\flat)$
and spanned by intersection points
of $C_i$ and $C_j$, to which we assign
the basis of $\mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_i,\ensuremath{\mathcal{E}}_j)$
as in Figure \ref{fg:vanishing_cycles}.
Notations for the basis of
$\mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_i, \ensuremath{\mathcal{E}}_j)$
is given in Appendix.
This defines the linear isomorphisms $\phi_{ij}$'s.
The commutativity
of (\ref{eq:diagram})
is verified
by counting triangles.
Let us illustrate this
with a few examples.
Take $x_1 \in \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_1, \ensuremath{\mathcal{E}}_4)$
and $x_1^\vee \in \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_4, \ensuremath{\mathcal{E}}_6)$,
whose composition is
$e_1 \in \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_1, \ensuremath{\mathcal{E}}_6)$.
On the Fukaya side,
we count the number of triangles
whose edges are contained in
$C_1 \cup C_4 \cup C_6$
and two of whose vertices are
$x_1$ and $x_1^\vee$.
Such a triangle exists uniquely,
whose remaining vertex is $e_1$
(Figure \ref{fg:cd1}).
\begin{figure}[H]
\centering
\psfrag{C3}{$C_1$}
\psfrag{C4}{$C_2$}
\psfrag{C5}{$C_3$}
\psfrag{C6}{$C_4$}
\psfrag{C7}{$C_5$}
\psfrag{C8}{$C_6$}
\psfrag{x1}{$x_1$}
\psfrag{x2}{$x_2$}
\psfrag{x3}{$x_3$}
\psfrag{x12*}{$x_1^\vee \wedge x_2^\vee$}
\psfrag{x23*}{$x_2^\vee \wedge x_3^\vee$}
\psfrag{x31*}{$x_3^\vee \wedge x_1^\vee$}
\psfrag{x1*}{$x_1^\vee$}
\psfrag{x2*}{$x_2^\vee$}
\psfrag{x3*}{$x_3^\vee$}
\psfrag{e1}{$e_{1}$}
\psfrag{e2}{$e_{2}$}
\psfrag{e3}{$e_{3}$}
\scalebox{0.7}{\includegraphics{cd1.eps}}
\caption{Counting triangle}
\label{fg:cd1}
\end{figure}

Next we try to compose
$x_3 \in \mathop{\mathrm{Hom}}\nolimits(C_3^\flat,C_4^\flat)$
and $x_1^\vee \in \mathop{\mathrm{Hom}}\nolimits(C_4^\flat, C_6^\flat)$
in the Fukaya category.
The only possibility
would be the triangle
with $e_3$ as the remaining vertex.
However, this is not allowed
because of
the missing point
(Figure \ref{fg:cd2}).

\begin{figure}[H]
\centering
\psfrag{C3}{$C_1$}
\psfrag{C4}{$C_2$}
\psfrag{C5}{$C_3$}
\psfrag{C6}{$C_4$}
\psfrag{C7}{$C_5$}
\psfrag{C8}{$C_6$}
\psfrag{x1}{$x_1$}
\psfrag{x2}{$x_2$}
\psfrag{x3}{$x_3$}
\psfrag{x12*}{$x_1^\vee \wedge x_2^\vee$}
\psfrag{x23*}{$x_2^\vee \wedge x_3^\vee$}
\psfrag{x31*}{$x_3^\vee \wedge x_1^\vee$}
\psfrag{x1*}{$x_1^\vee$}
\psfrag{x2*}{$x_2^\vee$}
\psfrag{x3*}{$x_3^\vee$}
\psfrag{e1}{$e_{1}$}
\psfrag{e2}{$e_{2}$}
\psfrag{e3}{$e_{3}$}
\scalebox{0.7}{\includegraphics{cd2.eps}}
\caption{No triangles}
\label{fg:cd2}
\end{figure}

This shows that
the composition of $x_3$
and $x_1^\vee$
in the Fukaya category
is zero,
which matches the table
in Appendix
calculated in $\ensuremath{{D^b \mathrm{coh}(Y)}}$.
Finally, as an example of
counting with signs,
we compute
$$
\mathop{\mathrm{Hom}}\nolimits(C_1^\flat, C_4^\flat) \times
   \mathop{\mathrm{Hom}}\nolimits(C_4^\flat,C_5^\flat)
 \ni (x_1, x_1^\vee \wedge x_2^\vee)
  \mapsto -x_2^\vee \in \mathop{\mathrm{Hom}}\nolimits(C_1^\flat, C_5^\flat).
$$
Indeed, the triangle
whose vertices are $x_1, x_1^\vee \wedge x_2^\vee$,
and $x_2^\vee$ contains one black dot
on its edge
(Figure \ref{fg:cd3}).
\begin{figure}[H]
\centering
\psfrag{C3}{$C_1$}
\psfrag{C4}{$C_2$}
\psfrag{C5}{$C_3$}
\psfrag{C6}{$C_4$}
\psfrag{C7}{$C_5$}
\psfrag{C8}{$C_6$}
\psfrag{x1}{$x_1$}
\psfrag{x2}{$x_2$}
\psfrag{x3}{$x_3$}
\psfrag{x12*}{$x_1^\vee \wedge x_2^\vee$}
\psfrag{x23*}{$x_2^\vee \wedge x_3^\vee$}
\psfrag{x31*}{$x_3^\vee \wedge x_1^\vee$}
\psfrag{x1*}{$x_1^\vee$}
\psfrag{x2*}{$x_2^\vee$}
\psfrag{x3*}{$x_3^\vee$}
\psfrag{e1}{$e_{1}$}
\psfrag{e2}{$e_{2}$}
\psfrag{e3}{$e_{3}$}
\scalebox{0.7}{\includegraphics{cd3.eps}}
\caption{Counting triangles with signs}
\label{fg:cd3}
\end{figure}
We can do similar analyses
for the rest of the triangles.
The result perfectly agrees with the computations
in $\ensuremath{{D^b \mathrm{coh}(Y)}}$.
\qed

Although the derived category of
an $A_\infty$-category
is usually defined
using twisted complexes,
we adopt the following definition
in this paper
since our $\Fuk^{\rightarrow} W$ satisfies
${\hom}_{\FukW}^k(C_i^\flat,C_j^\flat)=0$
for $k \neq 0$
and $\mathfrak{m}_k=0$
for $k \neq 2$.
\begin{definition}
Let $A=\oplus_{i,j}\Hom_{\FukW}(C_i^\flat,C_j^\flat)$
be the total morphism algebra.
The derived Fukaya category
$D^b \Fuk^{\rightarrow} W$ is
the bounded derived category
$D^b (\mathrm{mod}\mbox{-} A)$
of the category of right
finite-dimensional modules
over the algebra $A$.
\end{definition}
Now we can state our main theorem:
\begin{theorem} \label{th:main_theorem}
There exists an equivalence of triangulated categories
$$
\ensuremath{{D^b \mathrm{coh}(Y)}} \cong D^b \Fuk^{\rightarrow} W.
$$
\end{theorem}
\proof
From Theorem \ref{th:commutativity},
we have
$$
A \cong \oplus_{i,j} \mathop{\mathrm{Hom}}\nolimits_\ensuremath{{D^b \mathrm{coh}(Y)}}(\ensuremath{\mathcal{E}}_i,\ensuremath{\mathcal{E}}_j).
$$
Theorem \ref{th:main_theorem} follows
immediately from the theorem of Bondal
\cite{Bondal}
that $\ensuremath{{D^b \mathrm{coh}(Y)}} \cong D^b(\mathrm{mod}\mbox{-} (\oplus_{i,j} \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_i,\ensuremath{\mathcal{E}}_j)))$.
\qed

We can perform
similar analyses
for all the other toric del Pezzo surfaces
to obtain Theorem \ref{th:hms}.

\section{Appendix}

Here we give the table of compositions of morphisms
in the derived category of coherent sheaves
on the projective plane
$\ensuremath{\mathbb{P}}(V)$
blown-up at
$p_1 = [1:0:0]$, $p_2 = [0:1:0]$, $p_3 = [0:0:1]$.
We use the following notations:
$\ensuremath{\mathcal{E}}_i$'s are defined in (\ref{eq:ec}),
$\{x_i\}_{i=1}^3$ is the basis of $V$,
$\{x_i^\vee\}_{i=1}^3$ is the dual basis of $V^\vee$, and
$e_i$ is the generator of
$\mathop{\mathrm{Hom}}\nolimits(\ensuremath{\mathcal{E}}_i, \ensuremath{\mathcal{E}}_6)=\ensuremath{\mathbb{C}}$ for $i=1,2,3$.

$\mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_1}, \ensuremath{\scE_4})
 \times \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_4}, \ensuremath{\scE_5})
 \rightarrow \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_1}, \ensuremath{\scE_5})$
$$
\begin{array}{c|c|c|c}
 & x_1^\vee \wedge x_2^\vee
 & x_2^\vee \wedge x_3^\vee
 & x_3^\vee \wedge x_1^\vee \\
 \hline
 x_1 & -x_2^\vee & 0 & x_3^\vee
\end{array}
$$
$\mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_1}, \ensuremath{\scE_4}) \times \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_4}, \ensuremath{\scE_6})
 \rightarrow \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_1}, \ensuremath{\scE_6})$
$$
\begin{array}{c|c|c|c}
 & x_1^\vee & x_2^\vee & x_3^\vee \\
 \hline
 x_1 & e_1 & 0 & 0
\end{array}
$$
$\mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_1}, \ensuremath{\scE_5}) \times \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_5}, \ensuremath{\scE_6})
 \rightarrow \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_1}, \ensuremath{\scE_6})$
$$
\begin{array}{c|c|c|c}
 & x_1 & x_2 & x_3 \\
 \hline
 x_2^\vee & 0 & e_1 & 0 \\
 \hline
 x_3^\vee & 0 & 0 & e_1 \\
\end{array}
$$
$\mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_2}, \ensuremath{\scE_4}) \times \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_4}, \ensuremath{\scE_5})
 \rightarrow \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_2}, \ensuremath{\scE_5})$
$$
\begin{array}{c|c|c|c}
 & x_1^\vee \wedge x_2^\vee
 & x_2^\vee \wedge x_3^\vee
 & x_3^\vee \wedge x_1^\vee \\
 \hline
 x_2 & x_1^\vee & -x_3^\vee & 0
\end{array}
$$
$\mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_2}, \ensuremath{\scE_4}) \times \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_4}, \ensuremath{\scE_6})
 \rightarrow \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_2}, \ensuremath{\scE_6})$
$$
\begin{array}{c|c|c|c}
 & x_1^\vee & x_2^\vee & x_3^\vee \\
 \hline
 x_2 & 0 & e_2 & 0
\end{array}
$$
$\mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_2}, \ensuremath{\scE_5}) \times \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_5}, \ensuremath{\scE_6})
 \rightarrow \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_2}, \ensuremath{\scE_6})$
$$
\begin{array}{c|c|c|c}
 & x_1 & x_2 & x_3 \\
 \hline
 x_1^\vee & e_2 & 0 & 0 \\
 \hline
 x_3^\vee & 0 & 0 & e_2
\end{array}
$$
$\mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_3}, \ensuremath{\scE_4}) \times \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_4}, \ensuremath{\scE_5})
 \rightarrow \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_3}, \ensuremath{\scE_5})$
$$
\begin{array}{c|c|c|c}
 & x_1^\vee \wedge x_2^\vee
 & x_2^\vee \wedge x_3^\vee
 & x_3^\vee \wedge x_1^\vee \\
 \hline
 x_3 & 0 & x_2^\vee & -x_1^\vee
\end{array}
$$
$\mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_3}, \ensuremath{\scE_4}) \times \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_4}, \ensuremath{\scE_6})
 \rightarrow \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_3}, \ensuremath{\scE_6})$
$$
\begin{array}{c|c|c|c}
 & x_1^\vee & x_2^\vee & x_3^\vee \\
 \hline
 x_3 & 0 & 0 & e_3
\end{array}
$$
$\mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_3}, \ensuremath{\scE_5}) \times \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_5}, \ensuremath{\scE_6})
 \rightarrow \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_3}, \ensuremath{\scE_6})$
$$
\begin{array}{c|c|c|c}
 & x_1 & x_2 & x_3 \\
 \hline
 x_1^\vee & e_2 & 0 & 0 \\
 \hline
 x_2^\vee & 0 & e_3 & 0
\end{array}
$$
$\mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_4}, \ensuremath{\scE_5}) \times \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_5}, \ensuremath{\scE_6})
 \rightarrow \mathop{\mathrm{Hom}}\nolimits(\ensuremath{\scE_4}, \ensuremath{\scE_6})$
$$
\begin{array}{c|c|c|c}
 & x_1 & x_2 & x_3 \\
 \hline
 x_1^\vee \wedge x_2^\vee & x_2^\vee & -x_1^\vee & 0 \\
 \hline
 x_2^\vee \wedge x_3^\vee & 0 & x_3^\vee & -x_2^\vee \\
 \hline
 x_3^\vee \wedge x_1^\vee & -x_3^\vee & 0 & x_1^\vee \\
\end{array}
$$

\bibliographystyle{plain}


\section{Introduction}
\label{sec:intro}

In hierarchical galaxy formation models, based on the cold dark matter (CDM) paradigm,
very small dwarf galaxies are excellent candidates for earliest formed galaxies. Numerical
simulations based on these models  make definitive predictions about the average density
and shape of the dark matter distribution in  such galaxies. These dwarf
systems are typically dark matter dominated, even in the innermost  regions,
unlike bright galaxies where the stellar population is dynamically important.
Dwarf galaxies could hence provide an unique opportunity to compare observations
with the CDM simulations.

 According to the numerical  simulations, mass-density distribution
in the inner  parts of the simulated dark matter halos could be well
described by  a  cusp i.e.  r$^{-1}$ power law. This  cusp in the density
distribution  manifests  itself as a  steeply rising rotation curve in the
inner regions of galaxies. However, this prediction from CDM simulations
is found to disagree with the observed rotation curves of several
dwarf galaxies (e.g. deBlok \& Bosma 2002); the data indicate a constant
density core dark matter distribution. Apart from the shape of the dark
matter halos, numerical simulations also predict an anti-correlation
between the characteristic density and the virial mass of the dark matter
halos. In hierarchical scenarios, the  low mass halos form at the earlier times, when the
background was higher, hence dwarf  galaxies  should have larger dark
matter densities.

 All these  predictions of the numerical simulations were however untested
in the faintest dwarf galaxies, as it was widely believed that very faint dwarf
irregular galaxies do not show any systematic rotational motions.
From a systematic study of the kinematics of a sample of dwarfs, C\^{o}t\'{e}, Carignan
\& Freeman  (2000)  found that normal rotation is  seen only in galaxies
brighter than -14 mag, while fainter dwarfs have disturbed kinematics.
This result is consistent with the earlier findings of Lo, Sargent \& Young (1993), who
from a study of a sample of dwarfs  (with M$_{B\rm} \sim -9$ to M$_{B\rm} \sim -15$)
found that very faint dwarf irregulars have chaotic velocity fields. However, most of the
previous studies were done with a coarser velocity resolution and modest sensitivities.

Inorder to study the kinematics of very faint dwarf irregular galaxies, we obtained
high velocity resolution  and high sensitivity  GMRT HI 21 cm-line  observations of a sample
of galaxies with M$_B>-$12.5.
The GMRT has a hybrid configuration which simultaneously
provides both high angular resolution ($\sim$ 2$''$ if one uses baselines between the arm antennas)
as well as sensitivity to  extended emission (from baselines between the antennas in the central
square). This unique hybrid configuration of the GMRT makes it an excellent facility
to study such galaxies. The velocity resolution used for our observations was $\sim1.6$ kms$^{-1}$
and the  typical integration time on each source was $\sim 16~-$18 hrs, which gave  a  typical rms
of $\sim$ 1.0 mJy/Beam per channel. The galaxies in our sample have typical HI masses $\sim$ 10
$^{6-8}$ M$_\odot$.

We present here the results obtained from our GMRT observations.

\section{Results of The GMRT Observations}
\label{sec:result}

Figs.1[B],3[B] show the velocity fields for some of the galaxies in our sample.
Our GMRT observations clearly  show that the earlier conclusions about the kinematics
of faintest galaxies were a consequence of observational bias.  Since most of the
earlier  studies of faint dwarf galaxies were done with coarser
velocity resolutions ($\sim 6 - 7$ kms$^{-1}$) and modest sensitivities,
hence, such observations could not  detect systematic gradients, which
are typically $\sim 10-15$ kms$^{-1}$, in the velocity fields of such faint
dwarfs. On the other hand, our high velocity resolution and high
sensitivity observations found large scale systematic velocity gradients
in even the faintest galaxy in  our sample i.e.  DDO210. The velocity field for
 DDO210, differs significantly from the  velocity field derived earlier by
Lo et al.(1993). The pattern seen in the  GMRT velocity field of DDO210 (fig.~\ref{fig:camb_mom}[B]) is,
to zeroth order, consistent with that expected from rotation, on the other
hand, the velocity field derived by Lo et al.(1993) based on a coarser  velocity
resolution of $\sim$ 6 kms$^{-1}$, is indeed ``chaotic". This difference in the
observed kinematics suggests that high velocity resolution and high sensitivity
is  crucial in determining the systematic gradients in the velocity fields of
faint galaxies. For some of the galaxies in our sample this large scale
systematic gradients could be modeled as systematic rotation, hence allowing
us to derive the rotation curve for  those galaxies and to determine the structure
of their dark matter halos through mass modelling. The rest of this  paper discusses
the results obtained from the detailed mass modelling of two of our
sample galaxies viz. Camelopardalis B (Cam B;M$_B\sim-$10.9) and DDO210 (M$_B\sim-$10.6).

\begin{figure}[h!]
\rotatebox{-90}{\epsfig{file=GS75F1.PS,width=2.5in, height=5.5in}}
\caption{{\bf{[A]}}Integrated HI emission map of Cam B at
         $40^{''} \times 38^{''}$ resolution overlayed on the Digitised Sky Survey
	Image. The contour levels are $3.7, 8.8, 19.1, 24.3, 29.4, 34.6, 39.8, 44.9, 50.1,
	55.2, 60.4, 65.5~\&~ 70.7\times 10^{19}$ atoms~cm$^{-2}$.
{\bf{[B]}}The HI velocity field of Cam~B at $24^{''}\times 22^{''}$
          resolution. The contours are in steps of 1~kms$^{-1}$ and
          range from 70.0~kms$^{-1}$ (the extreme North East contour)
          to 84.0~kms$^{-1}$ (the extreme South West contour).
         }
\label{fig:camb_mom}
\end{figure}

\subsection{Camelopardalis B}
\label{ssec:camb}

Fig.~\ref{fig:camb_mom}[A] shows the integrated  HI column density image of Cam B at 40$''\times 38''$
resolution overlayed on the optical DSS image. The HI mass obtained from the integrated profile
(taking the distance to the galaxy to be 2.2 Mpc) is 5.3$\pm$0.5 $\times10^6$M$_\odot$ and the M$_{HI}$/L$_B$
ratio is found to be 1.4 in solar units.

 The velocity field derived from the moment analysis of 24$''\times 22''$ resolution data is shown
in the Fig.~\ref{fig:camb_mom}[B]. The velocity field is regular and the isovelocity contours are approximately
parallel, this is a signature of rigid body rotation. The kinematical major axis of the galaxy has a position
angle $\sim$ 215$^\circ$, i.e. it is well aligned with the major axis of both the HI distribution and the optical
emission.

\begin{figure}[h!]
\rotatebox{-90}{\epsfig{file=GS75F2.PS,width=3.0in,height=5.5in}}
\caption{{\bf{[A]}}The derived rotation curve for Cam B (dashes)
and the rotation curve after applying
          the asymmetric drift correction (dash dots).
{\bf{[B]}}Mass models for Cam B using the corrected rotation curve.
The points are the observed data. The total mass of gaseous disk (dashed line)
is $6.6\times10^6 M_\odot$ (after scaling the total HI mass by a factor of 1.25, to include
the contibution of primordial Helium). The stellar disk (short dash dot line) has
$\Upsilon_V=0.2$, giving a stellar mass of $0.7 \times10^6 M_\odot$. The
best fit total rotation curve for the constant density halo model is shown as
a solid line, while the contribution of the halo itself is shown as a
long dash dot line (the halo density is $\rho_0=13.7\times10^{-3}
M_\odot$ pc$^{-3}$). The best fit total rotation curve for an NFW type halo
(for $c=1.0$ and $\Upsilon_V=0.0$) is shown as a dotted line.
         }
\label{fig:camb_vrot}
\end{figure}

       The rotation curve for Cam B was derived  from the velocity fields at 40$''\times 38''$, 24$''\times 22''$
and 16$''\times 14''$ resolution using the tilted ring method (see Begum, Chengalur and Hopp 2003 for details). The
derived hybrid rotation curve is shown as a dashed curve in Fig.~\ref{fig:camb_vrot}[A]. We find that the peak
inclination corrected rotational velocity  for Cam B ($\sim$ 7.0 kms$^{-1}$) is comparable to the observed HI
dispersion i.e. V$_{\rm{max}}/\sigma_{\rm{HI}}\sim1.0$. This implies that the random motions provide significant
dynamical support to the system. In other words, the observed rotational velocities underestimate the total
dynamical mass in the galaxy due to a significant pressure support of  the HI gas. Hence,
 the observed rotation velocities were corrected for this pressure support, using the usual
``asymmetric drift" correction, before constructing mass models (see Begum et al (2003) for details).
The dot-dashed curve in Fig.~\ref{fig:camb_vrot}[A] shows the ``asymmetric drift" corrected rotation curve.

Using the ``asymmetric drift"  corrected curve, mass models for Cam B were derived.
The best fit mass model using a modified isothermal halo is shown
in Fig.~\ref{fig:camb_vrot}[B]. Also shown in the figure is
the best fit total rotation curve for an NFW type halo.
As can be seen, the kinematics of Cam~B is  well fit with
a modified isothermal halo while an NFW halo provides
a poor fit to the data.

 The ``asymmetric drift" corrected rotation curve for Cam B is
rising till the last measured point (Fig.~\ref{fig:camb_vrot}[B]), hence the core
radius of the isothermal halo could not be constrained
from the data. The best fit model for a constant density halo
gives central halo density ($\rho_0$) of 12.0$\times 10^{-3}~M_
\odot$~pc$^{-3}$.
The derived $\rho_0$ is relatively insensitive to the assumed
mass-to-light ratio of the stellar disk ($\Upsilon_V$). We found that
by changing  $\Upsilon_V$ from a value of 0 (minimum disk fit)
to a value of 2.0 (maximum disk fit), $\rho_0$ changes by $<$20\%.
  From the last measured point of the  observed rotation curve, a total dynamical mass of
1.1$\times 10^8 M_\odot$ is derived, i.e. at the last measured point
more than 90\% of the mass  of Cam~B is dark. Futher, the dominance of the
dark matter halo together with the linear shape of the rotation
curve (after correction for ``asymmetric drift'') means that one
cannot obtain a good fit to the rotation curve using an NFW halo
regardless of the assumed $\Upsilon_V$.

\begin{figure}[h!]
\rotatebox{0}{\epsfig{file=GS75F3.PS,width=5.5in}}
\caption{{\bf{[A]}} The optical DSS image of DDO210 (greyscales) with
         the GMRT 44$^{''}\times37^{''}$  resolution integrated HI
         emission map (contours) overlayed.
	The contour levels are $0.7, 10.2, 19.8, 29.3, 38.3, 45.5, 57.1,
	67.5, 77.1, 86.7, 96.2, 105.7, 121.4~ \& ~124.8 \times 10^{19}$ atoms~cm$^{-2}$.
{\bf{[B]}}The HI velocity field of DDO210 at 29$^{''}\times 23^{''}$
          resolution. The contours are in steps of 1~kms$^{-1}$ and
          range from $-$145.0~kms$^{-1}$ to $-$133.0~kms$^{-1}$.
         }
\label{fig:ddo210_mom}
\end{figure}

\begin{figure}[h!]
\rotatebox{-90}{\epsfig{file=GS75F4.PS,width=3.0in,height=5.5in}}
\caption{{\bf{[A]}}The hybrid rotation curve (dashes)  and the rotation curve
        after applying the asymmetric drift correction (dots).
{\bf{[B]}} Mass models for DDO210 using the corrected rotation curve.
         The points are the observed data. The total mass of gaseous
            disk (dashed line)  is $3.6\times10^6 M_\odot$ (including the contibution
	of primodial He).The stellar
          disk (short dash dot line) has
          $\Upsilon_B=3.4$, giving a stellar mass of $9.2 \times10^6 M_\odot$.
        The  best fit total rotation curve for the constant density
          halo model is shown as
          a solid line, while the contribution of the halo itself
           is shown as a  long dash dot line (the halo density is
           $\rho_0=29~\times10^{-3} M_\odot$ pc$^{-3}$).
         The best fit total rotation curve for an NFW type halo,
          using $\Upsilon_B=0.5$, c=5.0 and $v_{200}$=38.0~km s$^{-1}$
    is shown as a dotted line. See text for more details.
  }
\label{fig:ddo210_vrot}
\end{figure}

\subsection{DDO210}
\label{ssec:camb}

  DDO210 is the faintest known gas rich dwarf galaxy in our local group.
Karachentsev et al.(2002), based on the HST observations of the I magnitude of
the tip of the red giant branch, estimated the distance to this galaxy to
be 950$\pm$50 kpc.
Fig.~\ref{fig:ddo210_mom}[A] shows the integrated  HI column density image of
DDO210  at 44$''\times 37''$
resolution overlayed on the optical DSS image. The HI isodensity contours are
elongated in eastern and southern half of the galaxy, indicating a density enhancement
in these directions.The HI mass obtained from the integrated profile
(taking the distance to the galaxy to be 1.0 Mpc) is 2.8$\pm$0.3 $\times10^6$M$_\odot$
and the M$_{HI}$/L$_B$ ratio is found to be 1.0 in solar units.

 The velocity field of DDO210 derived from the moment analysis of 29$''\times 23''$
resolution data cube is shown in Fig.~\ref{fig:ddo210_mom}[B]. The velocity field is
regular and a systematic velocity gradient is seen across the galaxy.

 Fig.~\ref{fig:ddo210_vrot}[A] shows the derived rotation curve for DDO210 and the
``asymmetric drift" corrected rotation curve.  We find that the ``asymmetric
drift" corrected rotation curve of DDO210  can be well fit with  a
modified isothermal halo (with a central density $\rho_0 \sim 29\times10^
{-3}$ $M_\odot$ pc$^{-3}$). In the case of the NFW halo, we find that
there are a range of parameters which provide acceptable fits e.g. (v$_{200}\sim$ 20 kms$^{-1}$,
c $\sim$ 10) to (v$_{200}\sim$ 500 kms$^{-1}$, c$\sim$ 0.001); the halo parameters
could not be uniquely determined from the fit. However, the best fit value of the
concentration parameter c, at any given v$_{200}$ was found to be consistently smaller
than  the value predicted by numerical simulations for the $\Lambda$CDM universe (Bullock et al. 2001).
Fig.~\ref{fig:ddo210_vrot}[B] shows the best fit mass models for DDO210 (see Begum \& Chengalur 2004 for details).

\begin{figure}[h!]
\rotatebox{0}{\epsfig{file=GS75F5.PS,width=4.0in}}
\caption{Scatter plots of the central halo density against the
          circular velocity and the absolute blue magnitude . The data (empty squares)
          are from Verheijen~(1997), Broeils~(1992),
          C\^{o}t\'{e} et al.~(2000) and Swaters~(1999). The filled
          squares are the medians of the binned data, and the straight
          lines are the best fits to the data. Cam~B and DDO210 are
            shown  as crosses.
         }
\label{fig:dens}
\end{figure}

\section{Discussion}
\label{sec:result}

Fig.~\ref{fig:dens} shows  the core density of isothermal halo against circular
velocity and absolute blue magnitude for a sample of galaxies, spanning a range of
magnitudes from M$_B\sim-10.0$ mag to M$_B\sim-23.0$ mag. Cam B and DDO210 are also
shown in the figure, lying at the low luminosity end of the sample.
 As can be seen in the figure,
there is a trend of increasing halo density with a decrease in circular velocity
and absolute magnitude, shown by a best fit line to the data (solid line),
although the correlation is very weak and noisy. Further, as a guide to an
eye, we have also binned the data and plotted the median value (solid points).
Binned data also shows a similar trend. Such a tread is expected in
hierarchical structure formations scenario (e.g. Navarro,
Frenk \& White 1997).


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\begin{document}
\begin{CJK*}{GBK}{song}
\thispagestyle{empty}
\title{Freeze-Tag is NP-Hard in 3D with $L_1$ distance}

\author{Lucas de Oliveira Silva}

\vspace*{4mm}
\noindent {\small\bf Abstract} \quad  {\small Arkin et al. \cite{Arkin06} in 2002 introduced a scheduling-like problem called Freeze-Tag Problem (FTP) motivated by robot swarm activation. The input consists of the locations of $n$ mobile punctual robots in some metric space or graph. Only one begins ``active", while the others are initially ``frozen". All active robots can move at unit speed and, upon reaching a frozen one's location, activates it. The goal is to activate all the robots in the minimum amount of time, the so-called makespan. Until 2017 the hardness of this problem in metric spaces was still open, but then Yu et al. \cite{Yu17} proved it to be NP-Hard in the Euclidian plane, and in the same year, Demaine and Roudoy \cite{Erik17} demonstrated that the FTP is also hard in 3D with any $L_p$ distance (with $p>1$). However, we still don't know whether Demaine's and Roudoy's result could be translated to the plane. This paper fills the $p=1$ gap by showing that the FTP is NP-Hard in 3D with $L_1$ distance.}

\vspace*{3mm}
\noindent{\small\bf Keywords} \quad {\small Complexity, NP-Hardness, Scheduling, Computational Geometry, Swarm Robotics}
\vspace*{4mm}
\end{CJK*}

\baselineskip=18pt plus.2pt minus.2pt
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\begin{multicols}{2}

\section{Introduction}
The Freeze-Tag Problem was introduced by Arkin et al. in 2002 \cite{Arkin06} to model the automatic awakening of a robot swarm by manually turning on just one of the individuals. The input consists of a list of $n$ mobile punctual robots' locations in some metric space or graph. Only one of them begins ``active" (``on", ``unfrozen" or ``awake") and is called the source, while the others are initially ``frozen" (``off", ``frozen" or ``asleep"). Frozen robots become active when an active one reaches its location. After activated, a robot can move at unit speed and help unfreeze the remaining frozen ones. In the decision version, we are also given a time limit, and the goal is to decide whether all the robots can be activated within this limit.

Depending on the choice of metric (defining ``unit speed"), domain, or some other restriction, the problem takes different forms. For example, Arkin et al. \cite{Arkin06} proved that the FTP is NP-Hard on Weighted Star Graphs, but is in P when restricted to the unweighted ones. Moreover, by Bender et al. \cite{Arkin03}, we know that it is hard on Unweighted General Graphs. The first ones to tackle the NP-Hardness in metric spaces were Yu et al. at FWCG 2017 \cite{Yu17}, who proved the problem NP-Hard in the Euclidian plane, and later Demaine with Rudoy, that showed that the problem is also hard in 3D with any $L_p$ distance (with $p>1$). Our result complements the work of Demaine and Rudoy by proving that the FTP is also NP-Hard in 3D with $L_1$ distance.

\section{Reduction}
To prove the hardness claim, we reduce from $Monotone$-$3SAT$, shown to be NP-Complete in \cite{Go78}. An instance of this problem consists of a set of boolean variables $\{x_1, \dots, x_n\}$ and clauses $\{c_1, \dots, c_m\}$ in which three literals form each clause, and each literal is a, possibly negated, copy of one of the variables. Each clause is either \emph{positive} when all its literals are not negated or \emph{negative} when all are negated. The question is whether a satisfying boolean assignment to the variables exists, that is, an assignment such that every clause contains at least one true literal. Throughout this paper, Latin uppercase letters will denote points and the lowercase robots.

\begin{theorem}
Freeze-Tag is NP-Complete in 3D with $L_1$ distance.
\end{theorem}

\begin{proof}
As shown in \cite{Yu16}, the problem is in NP.

Given an instance $X=\{x_1, \dots, x_n\}$, $C=\{c_1, \dots, c_m\}$ (with $n>1$ and $m>2$) of the $Monotone$-$3SAT$ we can assume, without loss of generality, that $n$ is odd and the number of positive and negative clauses are the same. If not, we can add three dummy variables and a single positive clause containing them and then duplicate some clauses until the number of each type is equal. So let $C = \{c_1, \dots, c_{m/2}\} \cup \{\overline{c}_1, \dots, \overline{c}_{m/2}\}$ where the overline represents a negative clause. We place robots as follows:

Let $\alpha=\frac{2}{n-1}$, $\epsilon<\alpha/4$, $\beta=\frac{2}{m-2}$ and $L=6+\epsilon$. At the origin we have $n$ robots, $r_1, r_2, \dots, r_n$, with $r_1$ being the source one. For each variable $x_{i+1}$, there are two groups of robots, a \emph{true} one $t_{i+1}$ at $T_{i+1} = (2-i\alpha; i\alpha; \epsilon)$, and a false one $f_{i+1}$ at $F_{i+1} = (2-i\alpha; i\alpha; -\epsilon)$. The first corresponds to the occurrences of $x_{i+1}$ as a not negated literal, and the second corresponds to the negated ones. So each group contains a robot for each occurrence of $x_{i+1}$ in a clause being not negated or negated, respectively. The awakening order of these groups will represent the boolean value of $x_{i+1}$, that is, $x_{i+1} = true$ iff $t_{i+1}$ is awakened before $f_{i+1}$. Additionally, we add another $n$ robots, $s_1, s_2, \dots, s_n$, each $s_{i+1}$ being located at $S_{i+1} = (2-i\alpha+(L-2-4\epsilon)/2; i\alpha+(L-2-4\epsilon)/2; 0)$.

Then for each $k=0, \dots, m/2-1$ and each variable $x_{i+1}$ that does appear (not) negated in clause ($c_{k+1}$) $\overline{c}_{k+1}$, we create four robots:

$\mathbf{c}^{i+1}_{k+1}$ at $\mathbf{C}^{i+1}_{k+1} = T_{i+1} + (k\beta; k\beta; 2-2k\beta)$,

$\mathbf{d}^{i+1}_{k+1}$ at $\mathbf{D}^{i+1}_{k+1} = \mathbf{C}^{i+1}_{k+1} + P$,

$\overline{\mathbf{c}}^{i+1}_{k+1}$ at $\overline{\mathbf{C}}^{i+1}_{k+1} = F_{i+1} + (k\beta; k\beta; -2+2k\beta)$ and

$\overline{\mathbf{d}}^{i+1}_{k+1}$ at $\overline{\mathbf{D}}^{i+1}_{k+1} = \overline{\mathbf{C}}^{i+1}_{k+1} - P$,

where $P = (0; 0; L - 4 - 3\epsilon)$.

Finally, for each $k=0, \dots, m/2-1$ we add two robots:

$\mathbf{c}_{k+1}$ at $\mathbf{C}_{k+1} = C^{\ceil{n/2}}_{k+1} + Q$ and

$\overline{\mathbf{c}}_{k+1}$ at $\overline{\mathbf{C}}_{k+1} = \overline{C}^{\ceil{n/2}}_{k+1} + Q$, where $Q = (1; 1; 0)$.

We will prove that there is a satisfying boolean assignment to the given $Monotone$-$3SAT$ instance iff this FTP instance has an awakening schedule, with $r_1$ as the source, of optimal makespan $L$. \break

$(\implies)$ Suppose that the given $Monotone$-$3SAT$ has a satisfying boolean assignment. Without loss of generality, suppose that $T=\{x_1, \dots, x_l\}$ are the variables assigned to $true$ and $F=X\backslash T = \{x_{l+1}, \dots, x_n\}$ the variables assigned to $false$. Consider this scheduling:

At time instant zero, all $r_i$'s become awake. Then for each $r_i \in T$, $r_i$ goes to group $t_i$ at point $T_i$, and for each $r_j \in F$, $r_j$ goes to group $f_j$ at point $F_j$.

These robots reach their target at $2+\epsilon$. Upon arrival, a number of robots equal to the number of occurrences of the corresponding target literal become awake.
Afterward, the $r_i$'s go to their opposite still frozen group, and the other robots inside that first group split paths. Each one at a $T_i$ point goes to a different $\mathbf{c}_k^i$ robot that $x_i$ appears in the corresponding $c_k$ clause, and those at a $F_i$ point analogously go to a different $\overline{\mathbf{c}}^i_k$.

The $r_i$'s will reach their new destination group at $2+3\epsilon$ when another similar splitting occurs, now seeking to awaken those missing $\mathbf{c}_k^i$'s and $\overline{\mathbf{c}}^i_k$'s. They will be unfrozen at $4+3\epsilon$ and go to the matching $\mathbf{D}^i_k$ ($\overline{\mathbf{D}}^j_l$) point above (below) arriving in $L$. After this, each $r_i$ heads directly to a $s_i$ robot, getting there at time $L$.

In contrast, by $4+\epsilon$, the first batch of $\mathbf{c}^i_k$'s and $\overline{\mathbf{c}}^j_l$'s had already become awake and went directly to the corresponding $\mathbf{c}_k$ ($\overline{\mathbf{c}}_k$). Those who awakened them are now going to the matching $\mathbf{D}^i_k$ ($\overline{\mathbf{D}}^j_l$) point above (below), getting there at $L-2\epsilon$ when they'll finish.

At $4+\epsilon$, at least one robot will be heading each $\mathbf{c}_k$ ($\overline{\mathbf{c}}_k$) since the boolean assignment is a satisfying one. Furthermore, all these remaining robots are unfrozen by $L$.

$(\impliedby)$ Suppose the constructed FTP instance has an awakening schedule, with $r_1$ as the source, of optimal makespan $L$. The $n$ $s_i$'s are each at a distance $L-4\epsilon$ from the origin, and the distance between them is at least $\alpha>4\epsilon$. So, without loss of generality, we can assume that for each $i$, $r_i$ awakens $s_i$. While going to $s_i$, each $r_i$ has $4\epsilon$ time to spare. So there is an optimal solution where for each $i$, $r_i$ awakens the groups $t_i$ and $f_i$ before going to $s_i$. Consider that we are in this case.

Therefore we can construct a boolean assignment to the given $Monotone$-$3SAT$ instance as follows: $x_i = true$ iff $t_i$ is awakened before $f_i$. \break

Without loss of generality, after being awakened, each robot at a $t_i$ ($f_i$) group must go directly to a different $\mathbf{d}_k^i$ ($\overline{\mathbf{d}}^i_k$) robot that $x_i$ appears in the corresponding $c_k$ ($\overline{c}_k$) clause as there are as many of these as themselves. Because we are dealing with $L_1$ distance, they can also unfreeze the corresponding $\mathbf{c}_k^i$ or $\overline{\mathbf{c}}^i_k$ in their path. Also we can assume that they do so, as this is the earliest that they could've become unfroze anyway.

Every $\mathbf{c}_k^i$ ($\overline{\mathbf{c}}^i_k$) is at distance $2$ from its corresponding $\mathbf{c}_k$ ($\overline{\mathbf{c}}_k$). The literals that are set to \emph{true} awakens the robots corresponding to the causes that they appear on at time $4+\epsilon$, but the other $\mathbf{c}_k^i$ and  $\overline{\mathbf{c}}^i_k$ robots are reached only at $4+3\epsilon$. So only the first batch can reach its corresponding $\mathbf{c}_k$ ($\overline{\mathbf{c}}_k$) in time. We've assumed that we are dealing with optimal scheduling of makespan $L$, so each $\mathbf{c}_k$ ($\overline{\mathbf{c}}_k$) is awakened in the end by a literal whose boolean value is \emph{true}.

Therefore the boolean assignment makes each clause contain at least one \emph{true} literal, making it a satisfying boolean assignment. \break

By these previous implications, the FTP in 3D with $L_1$ distance is NP-Hard, so by the first claim, its NP-Complete.
\end{proof}

\section{Conclusion}
The obvious open problem is whether the result of this paper could also be applied to the 2D case. Considering that the Euclidian counterpart is NP-Hard, the FTP in 2D with $ L_1 $ distance may also be.

\small
\bibliographystyle{alpha}