Trichotomy for Integer Linear Systems Based on Their Sign Patterns

In this paper, we consider solving the integer linear systems, i.e., given a matrix A in R^{m*n}, a vector b in R^m, and a positive integer d, to compute an integer vector x in D^n such that Ax <= b, where m and n denote positive integers, R denotes the set of reals, and D={0,1,..., d-1}. The problem is one of the most fundamental NP-hard problems in computer science. For the problem, we propose a complexity index h which is based only on the sign pattern of A. For a real r, let ILS_=(r) denote the family of the problem instances I with h(I)=r. We then show the following trichotomy: - ILS_=(r) is linearly solvable, if r < 1, - ILS_=(r) is weakly NP-hard and pseudo-polynomially solvable, if r = 1, and - ILS_=(r) is strongly NP-hard, if r > 1. This, for example, includes the existing results that quadratic systems and Horn systems can be solved in pseudo-polynomial time

New Results on Cutting Plane Proofs for Horn Constraint Systems

In this paper, we investigate properties of cutting plane based refutations for a class of integer programs called Horn constraint systems (HCS). Briefly, a system of linear inequalities A * x >= b is called a Horn constraint system, if each entry in A belongs to the set {0,1,-1} and furthermore there is at most one positive entry per row. Our focus is on deriving refutations i.e., proofs of unsatisfiability of such programs using cutting planes as a proof system. We also look at several properties of these refutations. Horn constraint systems can be considered as a more general form of propositional Horn formulas, i.e., CNF formulas with at most one positive literal per clause. Cutting plane calculus (CP) is a well-known calculus for deciding the unsatisfiability of propositional CNF formulas and integer programs. Usually, CP consists of a pair of inference rules. These are called the addition rule (ADD) and the division rule (DIV). In this paper, we show that cutting plane calculus is still complete for Horn constraints when every intermediate constraint is required to be Horn. We also investigate the lengths of cutting plane proofs for Horn constraint systems

A Combinatorial Certifying Algorithm for Linear Programming Problems with Gainfree Leontief Substitution Systems

Linear programming (LP) problems with gainfree Leontief substitution systems have been intensively studied in economics and operations research, and include the feasibility problem of a class of Horn systems, which arises in, e.g., polyhedral combinatorics and logic. This subclass of LP problems admits a strongly polynomial time algorithm, where devising such an algorithm for general LP problems is one of the major theoretical open questions in mathematical optimization and computer science. Recently, much attention has been paid to devising certifying algorithms in software engineering, since those algorithms enable one to confirm the correctness of outputs of programs with simple computations. In this paper, we provide the first combinatorial (and strongly polynomial time) certifying algorithm for LP problems with gainfree Leontief substitution systems. As a by-product, we answer affirmatively an open question whether the feasibility problem of the class of Horn systems admits a combinatorial certifying algorithm

Analyzing Satisfiability and Refutability in Selected Constraint Systems

This dissertation is concerned with the satisfiability and refutability problems for several constraint systems. We examine both Boolean constraint systems, in which each variable is limited to the values true and false, and polyhedral constraint systems, in which each variable is limited to the set of real numbers R in the case of linear polyhedral systems or the set of integers Z in the case of integer polyhedral systems. An important aspect of our research is that we focus on providing certificates. That is, we provide satisfying assignments or easily checkable proofs of infeasibility depending on whether the instance is feasible or not. Providing easily checkable certificates has become a much sought after feature in algorithms, especially in light of spectacular failures in the implementations of some well-known algorithms. There exist a number of problems in the constraint-solving domain for which efficient algorithms have been proposed, but which lack a certifying counterpart. When examining Boolean constraint systems, we specifically look at systems of 2-CNF clauses and systems of Horn clauses. When examining polyhedral constraint systems, we specifically look at systems of difference constraints, systems of UTVPI constraints, and systems of Horn constraints. For each examined system, we determine several properties of general refutations and determine the complexity of finding restricted refutations. These restricted forms of refutation include read-once refutations, in which each constraint can be used at most once; literal-once refutations, in which for each literal at most one constraint containing that literal can be used; and unit refutations, in which each step of the refutation must use a constraint containing exactly one literal. The advantage of read-once refutations is that they are guaranteed to be short. Thus, while not every constraint system has a read-once refutation, the small size of the refutation guarantees easy checkability