On approximating optimal weight “no”-certificates in weighted difference constraint systems

Abstract

This paper is concerned with the design and analysis of approximation algorithms for the problem of determining the least weight refutation in a weighted difference constraint system. Recall that a difference constraint is a linear constraint of the form $x_{i}-x_{j} \le b_{ij}$ and a conjunction of such constraints is called a difference constraint system (DCS). In a weighted DCS (WDCS), a positive weight is associated with each constraint. Every infeasible constraint system has a refutation, which attests to its infeasibility. In the case of a DCS, this refutation is a subset of the input constraints, which when added together produces a contradiction of the form $0 \le -b$, $b> 0$. It follows that every refutation acts as a “no”-certificate. The length of a refutation is the number of constraints used in the derivation of a contradiction. Associated with a DCS $\mathbf{D: A\cdot x \le b}$ is its constraint network $\mathbf{G= \langle V,E, b \rangle }$. It is well-known that $\mathbf{D}$ is infeasible if and only if $\mathbf{G}$ contains a simple, negative cost cycle. Previous research has established that every negative cost cycle of length k in $\mathbf{G}$ corresponds exactly to a refutation of $\mathbf{D}$ using k constraints. It follows that the shortest refutation of $\mathbf{D}$ (i.e., the refutation which uses the fewest number of constraints) corresponds to the length of the shortest negative cycle in $\mathbf{G}$. The constraint network of a WDCS is represented by a constraint network $\mathbf{G = \langle V, E, b, l \rangle }$, where $\mathbf{l}:\mathbf{E \rightarrow \mathbb {N}}$ represents a function which associates a positive, integral length with each edge in $\mathbf{G}$. In the case of a WDCS, the weight of a refutation is defined as the sum of the lengths of the edges corresponding to the refutation. The problem of finding the minimum weight refutation in a WDCS is called the weighted optimal length resolution refutation (WOLRR) problem and is known to be NP-hard. In this paper, we describe a pseudo-polynomial time algorithm for the WOLRR problem and convert it into a fully polynomial time approximation scheme (FPTAS).