Computer Science

A sliding puzzle is a combination puzzle where a player slide pieces along certain routes on a board to reach a certain end-configuration. In this paper, we propose a novel measurement of complexity of massive sliding puzzles with paramodulation which is an inference method of automated reasoning. It turned out that by counting the number of clauses yielded with paramodulation, we can evaluate the difficulty of each puzzle. In experiment, we have generated 100 * 8 puzzles which passed the solvability checking by countering inversions. By doing this, we can distinguish the complexity of 8 puzzles with the number of generated with paramodulation. For example, board [2,3,6,1,7,8,5,4, hole] is the easiest with score 3008 and board [6,5,8,7,4,3,2,1, hole] is the most difficult with score 48653. Besides, we have succeeded to obverse several layers of complexity (the number of clauses generated) in 100 puzzles. We can conclude that proposal method can provide a new perspective of paramodulation complexity concerning sliding block puzzles.

In 1979 Valiant introduced the complexity class VNP of p-definable families of polynomials, he defined the reduction notion known as p-projection and he proved that the permanent polynomial and the Hamiltonian cycle polynomial are VNP-complete under p-projections. In 2001 Mulmuley and Sohoni (and independently Bürgisser) introduced the notion of border complexity to the study of the algebraic complexity of polynomials. In this algebraic machine model, instead of insisting on exact computation, approximations are allowed. This gives VNP the structure of a topological space. In this short note we study the set VNPC of VNP-complete polynomials. We show that the complement VNP \ VNPC lies dense in VNP. Quite surprisingly, we also prove that VNPC lies dense in VNP. We prove analogous statements for the complexity classes VF, VBP, and VP. The density of VNP \ VNPC holds for several different reduction notions: p-projections, border p-projections, c-reductions, and border c-reductions. We compare the relationship of the VNP-completeness notion under these reductions and separate most of the corresponding sets. Border reduction notions were introduced by Bringmann, Ikenmeyer, and Zuiddam (JACM 2018). Our paper is the first structured study of border reduction notions.

In this brief note, we prove that the existence of Nash equilibria on integer programming games is Σ p 2 -complete.

Recently, Cohen, Haeupler and Schulman gave an explicit construction of binary tree codes over polylogarithmic-sized output alphabet based on Pudlák's construction of maximum-distance-separable (MDS) tree codes using totally-non-singular triangular matrices. In this short note, we give a unified and simpler presentation of Pudlák and Cohen-Haeupler-Schulman's constructions.

We consider the Integer Linear Programming (ILP) problem max{ c ⊤ x:Ax≤b,x∈ Z n } , parameterized by a right-hand side vector b∈ Z m , where A∈ Z m×n is a matrix of the rank n . Let v be an optimal vertex of the Linear Programming (LP) relaxation max{ c ⊤ x:Ax≤b} and B be a corresponding optimal base. We show that, for almost all b∈ Z m , an optimal point of the square ILP problem max{ c ⊤ x: A B x≤ b B ,x∈ Z n } satisfies the constraints Ax≤b of the original problem. From works of R. Gomory it directly follows that the square ILP problem max{ c ⊤ x: A B x≤ b B ,x∈ Z n } can be solved by an algorithm of the arithmetic complexity O(n⋅δ⋅logδ) , where δ=|det A B | . Consequently, it can be shown that, for almost all b∈ Z m , the original problem max{ c ⊤ x:Ax≤b,x∈ Z n } can be solved by an algorithm of the arithmetic complexity O(n⋅Δ⋅logΔ) , where Δ is the maximum absolute value of n×n minors of A . By the same technique, we give new inequalities on the integrality gap and sparsity of a solution and slack variables. Another ingredient is a known lemma that states the equality of the maximum absolute values of rank minors of matrices with orthogonal columns. This lemma gives us a way to transform ILP problems of the type max{ c ⊤ x:Ax=b,x∈ Z n + } to problems of the previous type, here we assume that rank(A)=m and all the m×m minors of A are coprime. Consequently, it follows that, for almost all b∈ Z m , there exists an algorithm with the arithmetic complexity O((n−m)⋅Δ⋅logΔ) to solve the problem in the equality form. Sparsity and integrality gap bounds are also presented.

The paper presents a polynomial time approximation schema for the edge-weighted version of maximum k-vertex cover problem in bipartite graphs.

Recently, an interest in constructing pseudorandom or hitting set generators for restricted branching programs has increased, which is motivated by the fundamental issue of derandomizing space-bounded computations. Such constructions have been known only in the case of width 2 and in very restricted cases of bounded width. In this paper, we characterize the hitting sets for read-once branching programs of width 3 by a so-called richness condition. Namely, we show that such sets hit the class of read-once conjunctions of DNF and CNF (i.e. the weak richness). Moreover, we prove that any rich set extended with all strings within Hamming distance of 3 is a hitting set for read-once branching programs of width 3. Then, we show that any almost O(logn) -wise independent set satisfies the richness condition. By using such a set due to Alon et al. (1992) our result provides an explicit polynomial time construction of a hitting set for read-once branching programs of width 3 with acceptance probability ε>5/6 . We announced this result at conferences almost 10 years ago, including only proof sketches, which motivated a plenty of subsequent results on pseudorandom generators for restricted read-once branching programs. This paper contains our original detailed proof that has not been published yet.

After presentations of Raz and Tal's oracle separation of BQP and PH result, several people (e.g. Ryan O'Donnell, James Lee, Avishay Tal) suggested that the proof may be simplified by stochastic calculus. In this short note, we describe such a simplification.

It is discussed how the superstatistical formulation of effective Boltzmann factors can be related to the concept of Kolmogorov complexity, generating an infinite set of complexity measures (CMs) for quantifying information. At this level, the information is treated according to its background, which means that the CM depends on the inherent attributes of the information scenario. While the basic Boltzmann factor directly produces the standard complexity measure (SCM), it succeeds in the description of large-scale scenarios where the data components are not interrelated with themselves, thus adopting the behaviour of a gas. What happens in scenarios in which the presence of sources and sinks of information cannot be neglected, needs of a CM other than the one produced by the ordinary Boltzmann factor. We introduce a set of flexible CMs, without free parameters, that converge asymptotically to the Kolmogorov complexity, but also quantify the information in scenarios with a reasonable small density of states. We prove that these CMs are obtained from a generalised relative entropy and we suggest why such measures are the only compatible generalisations of the SCM.

Clique-width is one of the graph complexity measures leading to polynomial special-case algorithms for generally NP-complete problems, e.g. graph colourability. The best two currently known algorithms for verifying c-colourability of graphs represented as clique-width terms are optimised towards two different extreme cases, a constant number of colours and a very large number of colours. We present a way to unify these approaches in a single relatively simple algorithm that achieves the state of the art complexity in both cases. The unified algorithm also provides a speed-up for a large number of colours.