Finite problems and the logic of the weak law of excluded middle

A new characteristic of propositional formulas as operations on finite problems, the cardinality of a sufficient solution set, is defined. It is proved that if a formula is deducible in the logic of the weak law of excluded middle, then the cardinality of a sufficient solution set is bounded by a constant depending only on the number of variables; otherwise, the accessible cardinality of a sufficient solution set is close to (greater than the nth root of) its trivial upper bound. This statement is an analog of the authors result about the algorithmic complexity of sets obtained as values of propositional formulas, which was published previously. Also, we introduce the notion of Kolmogorov complexity of finite problems and obtain similar results.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 291–302.Original Russian Text Copyright © 2005 by A. V. Chernov.This revised version was published online in April 2005 with a corrected issue number.     

The polynomial bounds of proof complexity in Frege systems

The concept of a determinative set of variables for a propositional formula was introduced by one of the authors, which made it possible to distinguish the set of hard-determinable formulas. The proof complexity of a formula of this sort has exponential lower bounds in some proof systems of classical propositional calculus (cut-free sequent system, resolution system, analytic tableaux, cutting planes, and bounded Frege systems). In this paper we prove that the property of hard-determinability is insufficient for obtaining a superpolynomial lower bound of proof lines (sizes) in Frege systems: an example of a sequence of hard-determinable formulas is given whose proof complexities are polynomially bounded in every Frege system.     

On the use of suboptimal matchings for scaling and ordering sparse symmetric matrices

The use of matchings is a powerful technique for scaling and ordering sparse matrices prior to the solution of a linear system Ax = b. Traditional methods such as implemented by the HSL software package MC64 use the Hungarian algorithm to solve the maximum weight maximum cardinality matching problem. However, with advances in the algorithms and hardware used by direct methods for the parallelization of the factorization and solve phases, the serial Hungarian algorithm can represent an unacceptably large proportion of the total solution time for such solvers. Recently, auction algorithms and approximation algorithms have been suggested as alternatives for achieving near‐optimal solutions for the maximum weight maximum cardinality matching problem. In this paper, the efficacy of auction and approximation algorithms as replacements for the Hungarian algorithm is assessed in the context of sparse symmetric direct solvers when used in problems arising from a range of practical applications. High‐cardinality suboptimal matchings are shown to be as effective as optimal matchings for the purposes of scaling. However, matching‐based ordering techniques require that matchings are much closer to optimality before they become effective. The auction algorithm is demonstrated to be capable of finding such matchings significantly faster than the Hungarian algorithm, but our ‐approximation matching approach fails to consistently achieve a sufficient cardinality. Copyright © 2015 The Authors Numerical Linear Algebra with Applications Published by John Wiley & Sons Ltd.     

On sets of sets mutually intersecting in exactly one element

Let be a finite set, and letS be a set of subsets of mutually intersecting in exactly one element, such that all elements ofS have cardinality at mostr, and such that each element of is contained in at least two elements ofS.We give an upper bound for the cardinality of and a characterization of the pairs (,S) attaining the upper bound.The problem is equivalent to determining the maximum number of lines of a linear space having at mostr lines through a point.     

Embedding classical in minimal implicational logic

Consider the problem which set V of propositional variables suffices for whenever , where , and ?_c and ?_i denote derivability in classical and intuitionistic implicational logic, respectively. We give a direct proof that stability for the final propositional variable of the (implicational) formula A is sufficient; as a corollary one obtains Glivenko's theorem. Conversely, using Glivenko's theorem one can give an alternative proof of our result. As an alternative to stability we then consider the Peirce formula . It is an easy consequence of the result above that adding a single instance of the Peirce formula suffices to move from classical to intuitionistic derivability. Finally we consider the question whether one could do the same for minimal logic. Given a classical derivation of a propositional formula not involving ⊥, which instances of the Peirce formula suffice as additional premises to ensure derivability in minimal logic? We define a set of such Peirce formulas, and show that in general an unbounded number of them is necessary.     

Largest induced suborders satisfying the chain condition

For a finite ordered set P, let c(P) denote the cardinality of the largest subset Q such that the induced suborder on Q satisfies the Jordan-Dedekind chain condition (JDCC), i.e., every maximal chain in Q has the same cardinality. For positive integers n, let f(n) be the minimum of c(P) over all ordered sets P of cardinality n. We prove:      

Singular Antitone Systems

Given an ordered set P and an antitone map g : P P, we obtain necessary and sufficient conditions for the existence of an odd positive integer k such that g^k is isotone. The results obtained have a natural application to the dual space of an Ockham algebra. In particular, we determine the cardinality of the endomorphism semigroup of a finite subdirectly irreducible Ockham algebra.     

Algorithms for clique-independent sets on subclasses of circular-arc graphs

A circular-arc graph is the intersection graph of arcs on a circle. A Helly circular-arc graph is a circular-arc graph admitting a model whose arcs satisfy the Helly property. A clique-independent set of a graph is a set of pairwise disjoint cliques of the graph. It is NP-hard to compute the maximum cardinality of a clique-independent set for a general graph. In the present paper, we propose polynomial time algorithms for finding the maximum cardinality and weight of a clique-independent set of a -free CA graph. Also, we apply the algorithms to the special case of an HCA graph. The complexity of the proposed algorithm for the cardinality problem in HCA graphs is O(n). This represents an improvement over the existing algorithm by Guruswami and Pandu Rangan, whose complexity is O(n²). These algorithms suppose that an HCA model of the graph is given.     

On the heat flow equation of surfaces of constant mean curvature in higher dimensions

In this paper, we consider the heat flow for the Hsystem with constant mean curvature in higher dimensions. We give sufficient conditions on the initial data such that the heat flow develops finite time singularity. We also provide a new set of initial data to guarantee the existence of global regular solution to the heat flow, that converges to zero in W 1,n with the decay rate t 2/(2-n) as time goes to infinity.     

On Finite Difference Sets

We establish a lower estimate of the cardinality of finite difference sets in the three dimensional Euclidean space R3 by showing that ¦A - A¦ 4.5 ¦A¦ -9 for every three dimensional finite set A. In addition, we describe the structure of two and three dimensional sets with the smallest cardinality of the difference set.     

Superlinearly Convergent Infeasible-Interior-Point Algorithm for Degenerate LCP

A large-step infeasible path-following method is proposed for solving general linear complementarity problems with sufficient matrices. If the problem has a solution, the algorithm is superlinearly convergent from any positive starting points, even for degenerate problems. The algorithm generates points in a large neighborhood of the central path. Each iteration requires only one matrix factorization and at most three (asymptotically only two) backsolves. It has been recently proved that any sufficient matrix is a P _*()-matrix for some 0. The computational complexity of the algorithm depends on as well as on a feasibility measure of the starting point. If the starting point is feasible or close to being feasible, then the iteration complexity is . Otherwise, for arbitrary positive and large enough starting points, the iteration complexity is O((1 + )² nL). We note that, while computational complexity depends on , the algorithm itself does not.     

Finding induced trees

Let G=(V(G),E(G)) be a finite connected undirected graph and WV(G) a subset of vertices. We are searching for a subset XV(G) such that WX and the subgraph induced on X is a tree. -completeness results and polynomial time algorithms are given assuming that the cardinality of W is fixed or not. Besides we give complexity results when X has to induce a path or when G is a weighted graph. We also consider problems where the cardinality of X has to be minimized.     

Complementary Sets of Finite Sets

Subsets 𝒜, 𝒮 of an additive group G are complementary if 𝒜 + 𝒮 = G. When 𝒜 is of finite cardinality ∣𝒜∣, and G is ℤ or ℝ, we give sufficient conditions for the existence of a complementary set 𝒮 with “density” not much larger than 1/∣𝒜∣. Supported in part by NSF DMS-0074531. Received February 14, 2002; in revised form July 18, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Antichains of Bounded Size in the Class of Tournaments

Tournament embedding is an order relation on the class of finite tournaments. An antichain is a set of finite tournaments that are pairwise incomparable in this ordering. We say an antichain can be extended to an antichain . Those finite antichains that can not be extended to antichains of arbitrarily large finite cardinality are exactly those that contain a member of each of four families of tournaments. We give an upper bound on the cardinality of extensions of such antichains. This bound depends on the maximum order of the tournaments in the antichain. This revised version was published online in June 2006 with corrections to the Cover Date.     

Small Point Sets that Meet All Generators of W(2n+1,q)

Let W(2n+1,q), n1, be the symplectic polar space of finite order q and (projective) rank n. We investigate the smallest cardinality of a set of points that meets every generator of W(2n+1,q). For q even, we show that this cardinality is q ⁿ⁺¹+q ^{n–1, and we characterize all sets of this cardinality. For q odd, better bounds are known.     

An application of the greedy heuristic of set cover to traffic checks

Let us suppose that X is a given, finite, not empty set and ${\mathcal F}Let us suppose that X is a given, finite, not empty set and is a given collection of subsets of X such that their union equals X, in other words covers X. Set cover is the problem of selecting as few as possible subsets from such that their union covers X. Max k-cover is the problem of selecting k subsets from such that their union has maximum cardinality. Both problems are NP-hard. There is a polynomial time greedy heuristic that iteratively selects the subset from that covers the largest number of yet uncovered elements. We implemented this greedy algorithm to support the planning of a checking system that is aimed to check the vehicles in a road network. We would like to answer such questions:

–	How many and which links are sufficient to check a given percentage of all traffic flow?
–	What percentage of traffic can be checked with given links?

This paper defines the necessary data and basic knowledge, gives algorithms to answer the previous questions and also shows the results of an implementation in a road network that contains about 11,000 junctions, 3,000 origin–destination junctions and 26,000 links.     

An exact algorithm for the maximum stable set problem

We describe a new branch-and-bound algorithm for the exact solution of the maximum cardinality stable set problem. The bounding phase is based on a variation of the standard greedy algorithm for finding a colouring of a graph. Two different node-fixing heuristics are also described. Computational tests on random and structured graphs and very large graphs corresponding to real-life problems show that the algorithm is competitive with the fastest algorithms known so far.This work has been supported by Agenzia Spaziale Italiana.     

A Polynomial Cutting Surfaces Algorithm for the Convex Feasibility Problem Defined by Self-Concordant Inequalities

Consider a nonempty convex set in ^m which is defined by a finite number of smooth convex inequalities and which admits a self-concordant logarithmic barrier. We study the analytic center based column generation algorithm for the problem of finding a feasible point in this set. At each iteration the algorithm computes an approximate analytic center of the set defined by the inequalities generated in the previous iterations. If this approximate analytic center is a solution, then the algorithm terminates; otherwise either an existing inequality is shifted or a new inequality is added into the system. As the number of iterations increases, the set defined by the generated inequalities shrinks and the algorithm eventually finds a solution of the problem. The algorithm can be thought of as an extension of the classical cutting plane method. The difference is that we use analytic centers and convex cuts instead of arbitrary infeasible points and linear cuts. In contrast to the cutting plane method, the algorithm has a polynomial worst case complexity of O(Nlog 1/) on the total number of cuts to be used, where N is the number of convex inequalities in the original problem and is the maximum common slack of the original inequality system.     

Representing posets with k-tournaments

In this paper, we introduce a binary relation on the vertex set of a k-tournament, and using this relation show that every finite poset with cardinality n4 can be represented by a k-tournament for every k satisfying 3kn–1.     

On Center Cycles in Grid Graphs

Finding good cycles in graphs is a problem of great interest in graph theory as well as in locational analysis. We show that the center and median problems are NP-hard in general graphs. This result holds both for the variable cardinality case (i.e., all cycles of the graph are considered) and the fixed cardinality case (i.e., only cycles with a given cardinality p are feasible). Hence it is of interest to investigate special cases where the problem is solvable in polynomial time. In grid graphs, the variable cardinality case is, for instance, trivially solvable if the shape of the cycle can be chosen freely. If the shape is fixed to be a rectangle one can analyze rectangles in grid graphs with, in sequence, fixed dimension, fixed cardinality, and variable cardinality. In all cases a complete characterization of the optimal cycles and closed form expressions of the optimal objective values are given, yielding polynomial time algorithms for all cases of center rectangle problems. Finally, it is shown that center cycles can be chosen as rectangles for bounded cardinalities such that the center cycle problem in grid graphs is in these cases completely solved.