Optimal Length Tree-Like Refutations of Linear Feasibility in UTVPI Constraints

Abstract

In this paper, we propose two algorithms for determining the optimal length tree-like refutation of linear feasibility in Unit Two Variable Per Inequality (UTVPI) constraints. Given an infeasible UTVPI constraint system (UCS), a refutation certifies its infeasibility. The problem of finding refutations in a UCS finds applications in domains such as program verification and operations research. In general, there exist several types of refutations of feasibility in constraint systems. In this paper, we focus on a specific type of refutation called a tree-like refutation. Tree-like refutations are complete, in that if a system of linear constraints is infeasible, then it must have a tree-like refutation. Associated with a refutation is its length which corresponds to the total number of constraints (including repeats) that are used to establish the infeasibility of the corresponding linear constraint system. Our goal in this paper is to find the optimal (minimum) length tree-like refutation (OTLR) of an infeasible UCS. We show that an OTLR of a UCS can be found in \(O(m \cdot n \cdot k)\) time, where m is the number of constraints, n is the number of variables in the system, and k is the length of an OTLR. We also propose a true-biased, randomized algorithm for this problem. This algorithm runs in \(O(m\cdot n \cdot \log n)\) time, and returns an OTLR with probability \((1-\frac{1}{e})\).