Polynomial time algorithms for optimal length tree-like refutations of linear infeasibility in UTVPI constraints

Abstract In this paper, we propose several algorithms for determining an optimal length tree-like refutation of linear feasibility in systems of 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 the sense 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 equals the total number of constraints (accounting for the reuse of constraints) that are used to establish the infeasibility of the corresponding linear constraint system. Our goal in this paper is to find an optimal length tree-like refutation (OTLR) of an infeasible UCS. We describe two deterministic algorithms that find an OTLR of a UCS. If m is the number of constraints, n is the number of variables in the system, and k is the length of an OTLR, then our two algorithms run in O ( n 3 ⋅ log k ) time and O ( m ⋅ n ⋅ k ) time respectively. We also propose a true-biased, randomized algorithm for this problem. This algorithm runs in O ( m ⋅ n ⋅ log n ) time, and returns the length of an OTLR with probability ( 1 − 1 e ) . We extend our work to weighted UTVPI constraint systems (WUCS), where a positive weight is associated with each UTVPI constraint. The weight of a tree-like refutation is defined as the sum of the weights of the constraints used by the refutation. The problem of finding a minimum weight refutation in a WUCS is called the weighted optimal length tree-like refutation (WOTLR) problem and is known to be NP-hard. We propose a pseudo-polynomial time algorithm for the WOTLR problem and convert it into a fully polynomial time approximation scheme (FPTAS).