47553632
On the Theory of Matchgate Computations
Valiant has proposed a new theory of algorithmic computation based on perfect matchings and Pfaffians. We study the properties of matchgates - the basic building blocks in this new theory. We give a set of algebraic identities which completely characterizes these objects for arbitrary numbers of inputs and outputs. These identities are derived from Grassmann-Plucker identities. The 4 by 4 matchgate character matrices are of particular interest. These were used in Valiant's classical simulation of a fragment of quantum computations. For these 4 by 4 matchgates, we use Jacobi's theorem on compound matrices to prove that the invertible matchgate matrices form a multiplicative group. Our results can also be expressed in the theory of holographic algorithms in terms of realizable standard signatures. These results are useful in establishing limitations on the ultimate capabilities of Valiant's theory of matchgate computations and holographic algorithms.
1980209
Basis Collapse in Holographic Algorithms
Abstract.Holographic algorithms are a novel approach to design polynomial time computations using linear superpositions. Most holographic algorithms are designed with basis vectors of dimension 2. Recently Valiant showed that a basis of dimension 4 can be used to solve in P an interesting (restrictive SAT) counting problem mod 7. This problem without modulo 7 is #P-complete, and counting mod 2 is NP-hard.We give a general collapse theorem for bases of dimension 4 to dimension 2 in the holographic algorithms framework. We also define an extension of holographic algorithms to allow more general support vectors. Finally we give a Basis Folding Theorem showing that in a natural setting the support vectors can be simulated by bases of dimension 2.
43665408
Resolution of Hartmanis' Conjecture for NL-Hard Sparse Sets
We resolve a conjecture of Hartmanis from 1978 about sparse hard sets for nondeterministic logspace (NL). We show that there exists a sparse hard set S for NL under logspace many-one reductions if and only if NL = L (deterministic logspace).
11342779
On Proving Circuit Lower Bounds against the Polynomial-Time Hierarchy: Positive and Negative Results
We consider the problem of proving circuit lower bounds against the polynomial-time hierarchy. We give both positive and negative results. For the positive side, for any fixed integer k > 0, we give an explicit Σ2p language, acceptable by a Σ2p-machine with running time O(nk2+k), that requires circuit size > nk. For the negative side, we propose a new stringent notion of relativization, and prove under this stringent relativization that every language in the polynomial-time hierarchy has polynomial circuit size. (For technical details, see also [CW03].)
16802575
On the Hardness of Permanent
We prove that if there is a polynomial time algorithm which computes the permanent of a matrix of order n for any inverse polynomial fraction of all inputs, then there is a BPP algorithm computing the permanent for every matrix. It follows that this hypothesis implies P #P = BPP. Our algorithm works over any sufficiently large finite field (polynomially larger than the inverse of the assumed success ratio), or any interval of integers of similar range. The assumed algorithm can also be a probabilistic polynomial time algorithm. Our result is essentially the best possible based on any black box assumption of permanent solvers, and is a simultaneous improvement of the results of Gemmell and Sudan [GS92], Feige and Lund [FL92] as well as Cai and Hemachandra [CH91], and Toda (see [ABG90]).
41950213
Holographic algorithms: From art to science
We develop the theory of holographic algorithms initiated by Leslie Valiant. First we define a basis manifold. Then we characterize algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and give a polynomial time decision algorithm for the simultaneous realizability problem. These results enable one to decide whether suitable signatures for a holographic algorithm are realizable, and if so, to find a suitable linear basis to realize these signatures by an efficient algorithm. Using the general machinery we are able to give unexpected holographic algorithms for some counting problems, modulo certain Mersenne type integers. These counting problems are #P-complete without the moduli. Going beyond symmetric signatures, we define d-admissibility and d-realizability for general signatures, and give a characterization of 2-admissibility and some general constructions of admissible and realizable families.
12735579
The enumeration of rooted nonseparable nearly cubic maps
Abstract This paper provides the functional equations satisfied by the generating functions for enumerating rooted nonseparable near-cubic planar maps with the size and the valency of root-vertex and/or root-face of the maps as parameters. Moreover, explicit expressions of these functions are also derived. Two of them are summation-free.
44495329
Enumeration on Nonseparable Planar Maps
Abstract This paper provides some functional equations satisfied by the generating functions for nonseparable rooted planar maps with the valency of root-vertex, the number of edges and the valency of root-faces of the maps as three parameters. But the solutions of these equations can only be obtained indirectly by considering some relations between nonseparable and general rooted planar maps. One of them is an answer to the open problem 6.1 in Liu (1983, Comb. Optim. CORR83-26, University of Waterloo).
38252080
A New Transference Theorem in the Geometry of Numbers
We prove a new transference theorem in the geometry of numbers, giving optimal bounds relating the successive minima of a lattice with the minimal length of generating vectors of its dual. It generalizes the transference theorem due to Banaszczyk. The theorem is motivated by our efforts to improve Ajtai's connection factors in the connection of average-case to worst-case complexity of the shortest lattice vector problem. Our proofs are non-constructive, based on methods from harmonic analysis.
31469931
STARS: A Socio-Technical Framework for Integrating Design Knowledge over the Internet
Traditional design tools usually model collaborative work as a process of data access and sharing. In globally distributed environments, however, it becomes important to model the interdisciplinary perspectives that individual contributors bring to the data. We refer to the individuals involved in the design process as stakeholders. Each stakeholder comes to the project with a different perspective: that is, different preferences, environments, and knowledge. An information system that accounts for these social and technical factors would not only manipulate design data, but also facilitate the reconciliation of these design perspectives over the Internet. This article presents a conceptual framework for supporting knowledge integration by modeling stakeholders' design perspectives. We describe the Social-Technical Analysis Research System (STARS), a prototype implementation system that uses advanced networking techniques to support stakeholders' interaction in collaborative design.