12359075
Geometric knapsack problems
We study a variety of geometric versions of the classical knapsack problem. In particular, we consider the following “fence enclosure” problem: given a setS ofn points in the plane with valuesvi ≥ 0, we wish to enclose a subset of the points with a fence (a simple closed curve) in order to maximize the “value” of the enclosure. The value of the enclosure is defined to be the sum of the values of the enclosed points minus the cost of the fence. We consider various versions of the problem, such as allowingS to consist of points and/or simple polygons. Other versions of the problems are obtained by restricting the total amount of fence available and also allowing the enclosure to consist of at mostM connected components. When there is an upper bound on the length of fence available, we show that the problem is NP-complete. We also provide polynomial-time algorithms for many versions of the fence problem when an unrestricted amount of fence is available.
11557966
An algorithmic study of manufacturing paperclips and other folded structures
We study algorithmic aspects of bending wires and sheet metal into a specified structure. Problems of this type are closely related to the question of deciding whether a simple non-self-intersecting wire structure (a carpenter's ruler) can be straightened, a problem that was open for several years and has only recently been solved in the affirmative. If we impose some of the constraints that are imposed by the manufacturing process, we obtain quite different results. In particular, we study the variant of the carpenter's ruler problem in which there is a restriction that only one joint can be modified at a time. For a linkage that does not self-intersect or self-touch, the recent results of Connelly et al. and Streinu imply that it can always be straightened, modifying one joint at a time. However, we show that for a linkage with even a single vertex degeneracy, it becomes NP-hard to decide if it can be straightened while altering only one joint at a time. If we add the restriction that each joint can be altered at most once, we show that the problem is NP-complete even without vertex degeneracies. In the special case, arising in wire forming manufacturing, that each joint can be altered at most once, and must be done sequentially from one or both ends of the linkage, we give an efficient algorithm to determine if a linkage can be straightened.
16007270
Point Probe Decision Trees for Geometric Concept Classes
A fundamental problem in model-based computer vision is that of identifying to which of a given set of concept classes of geometric models an observed model belongs. Considering a “probe” to be an oracle that tells whether or not the observed model is present at a given point in an image, we study the problem of computing efficient strategies (“decision trees”) for probing an image, with the goal to minimize the number of probes necessary (in the worst case) to determine in which class the observed model belongs. We prove a hardness result and give strategies that obtain decision trees whose height is within a log factor of optimal.
207132060
Terrain Decomposition and Layered Manufacturing
We consider a problem that arises in generating three-dimensional models by methods of layered manufacturing: How does one decompose a given model P into a small number of sub-models each of which is a terrain polyhedron? Terrain polyhedra have a base facet such that, for each point of the polyhedron, the line segment joining the point to its orthogonal projection on the base facet lies within the polyhedron. Terrain polyhedra are exactly the class of polyhedral models for which it is possible to construct the model using layered manufacturing (with layers parallel to the base facet), without the need for constructing "supports" (which must later be removed). In order to maximize the integrity of a prototype, one wants to minimize the number of individual sub-models that are manufactured and then glued together. We show that it is NP-hard to decide if a three-dimensional model P of genus 0 can be decomposed into k terrain polyhedra. We also prove a two-dimensional version of this theorem, for the case in which P is a polygonal region with holes. Both results still hold if we are restricted to isothetic objects and/or axis-parallel layering directions.
7753820
Approximate Geometric Pattern Matching Under Rigid Motions
We present techniques for matching point-sets in two and three dimensions under rigid-body transformations. We prove bounds on the worst-case performance of these algorithms to be within a small constant factor of optimal and conduct experiments to show that the average performance of these matching algorithms is often better than that predicted by the worst-case bounds.
57900050
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational geometry is presented.
2523975
Multiset rewriting and the complexity of bounded security protocols
We formalize the Dolev-Yao model of security protocols, using a notation based on multiset rewriting with existentials. The goals are to provide a simple formal notation for describing security protocols, to formalize the assumptions of the Dolev-Yao model using this notation, and to analyze the complexity of the secrecy problem under various restrictions. We prove that, even for the case where we restrict the size of messages and the depth of message encryption, the secrecy problem is undecidable for the case of an unrestricted number of protocol roles and an unbounded number of new nonces. We also identify several decidable classes, including a DEXP-complete class when the number of nonces is restricted, and an NP-complete class when both the number of nonces and the number of roles is restricted. We point out a remaining open complexity problem, and discuss the implications these results have on the general topic of protocol analysis.
5553551
Decision Problems for Propositional Linear Logic
Abstract Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. This accounting is made possible by removing the ‘structural’ rules of contraction and weakening, adding a modal operator and adding finer versions of the propositional connectives. Linear logic has fundamental logical interest and applications to computer science, particularly to Petri nets, concurrency, storage allocation, garbage collection and the control structure of logic programs. In addition, there is a direct correspondence between polynomial-time computation and proof normalization in a bounded form of linear logic. In this paper we show that unlike most other propositional (quantifier-free) logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, which indicates unboundedness of resources, the decision problem becomes PSPACE-complete. We also establish membership in NP for the multiplicative fragment, NP-completeness for the multiplicative fragment extended with unrestricted weakening, and undecidability for fragments of noncommutative propositional linear logic.
12005506
Kripke-Style Models for Typed lambda Calculus
The semantics of typed lambda calculus is usually described using Henkin models, consisting of functions over some collection of sets, or concrete cartesian closed categories, which are essentially equivalent. We describe a more general class of Kripte-style models. In categorical terms, our Kripke lambda models are cartesian closed subcategories of the presheaves over a poset. While the traditional lambda calculus proof system is not complete for Henkin models that may have empty types, we prove strong completeness for Kripke models. In fact, every set of equations that is closed under implication is the theory of a single Kripke model. We develop some properties of logical relations over Kripke structures, showing that every theory if the theory of a model determined by a Kripke equivalence relation over a Henkin model
26562844
Understanding SPKI/SDSI using first-order logic
SPKI/SDSI is a language for expressing distributed access control policy, derived from SPKI and SDSI. We provide a first-order logic (FOL) semantics for SDSI, and show that it has several advantages over previous semantics. For example, the FOL semantics is easily extended to additional policy concepts and gives meaning to a larger class of access control and other policy analysis queries. We prove that the FOL semantics is equivalent to the string rewriting semantics used by SDSI designers, for all queries associated with the rewriting semantics. We also provide a FOL semantics for SPKI/SDSI. This reveals some problems. For example, the standard proof procedure in RFC 2693 is semantically incomplete. In addition, as noted before by other authors, authorization tags in SPKI/SDSI are algorithmically problematic, making a complete proof procedure unlikely. We compare SPKI/SDSI with RT/sub 1//sup C/, which is a language in the RT role-based trust-management framework that can be viewed as an extension of SDSI. The constraint feature of /sub 1//sup C/, based on constraint datalog, provides an alternative mechanism that is expressively similar to SPKI/SDSI tags, semantically natural, and algorithmically tractable.
1053621
Privacy and contextual integrity: framework and applications
Contextual integrity is a conceptual framework for understanding privacy expectations and their implications developed in the literature on law, public policy, and political philosophy. We formalize some aspects of contextual integrity in a logical framework for expressing and reasoning about norms of transmission of personal information. In comparison with access control and privacy policy frameworks such as RBAC, EPAL, and P3P, these norms focus on who personal information is about, how it is transmitted, and past and future actions by both the subject and the users of the information. Norms can be positive or negative depending on whether they refer to actions that are allowed or disallowed. Our model is expressive enough to capture naturally many notions of privacy found in legislation, including those found in HIPAA, COPPA, and GLBA. A number of important problems regarding compliance with privacy norms, future requirements associated with specific actions, and relations between policies and legal standards reduce to standard decision procedures for temporal logic