Improved algorithms for optimal length resolution refutation in difference constraint systems

This paper is concerned with the design and analysis of improved algorithms for determining the optimal length resolution refutation (OLRR) of a system of difference constraints over an integral domain. The problem of finding short explanations for unsatisfiable Difference Constraint Systems (DCS) finds applications in a number of design domains including program verification, proof theory, real-time scheduling, and operations research. These explanations have also been called “certificates” and “refutations” in the literature. This problem was first studied in Subramani (J Autom Reason 43(2):121–137, 2009), wherein the first polynomial time algorithm was proposed. In this paper, we propose two new strongly polynomial algorithms which improve on the existing time bound. Our first algorithm, which we call the edge progression approach, runs in O(n2 · k + m · n · k) time, while our second algorithm, which we call the edge relaxation approach, runs in O(m · n · k) time, where m is the number of constraints in the DCS, n is the number of program variables, and k denotes the length of the shortest refutation. We conducted an extensive empirical analysis of the three OLRR algorithms discussed in this paper. Our experiments indicate that in the case of sparse graphs, the new algorithms discussed in this paper are superior to the algorithm in Subramani (J Autom Reason 43(2):121–137, 2009). Likewise, in the case of dense graphs, the approach in Subramani (J Autom Reason 43(2):121–137, 2009) is superior to the algorithms described in this paper. One surprising observation is the superiority of the edge relaxation algorithm over the edge progression algorithm in all cases, although both algorithms have the same asymptotic time complexity.