Algorithms and Complexity of Difference Logic

Abstract

Difference Logic (DL) is a fragment of linear arithmetics where atoms are constraints x+k <= y for variables x,y (ranging over Q or Z) and integer k. We study the complexity of deciding the truth of existential DL sentences. This problem appears in many contexts: examples include verification, bioinformatics, telecommunications, and spatio-temporal reasoning in AI. We begin by considering sentences in CNF with rational-valued variables. We restrict the allowed clauses via two natural parameters: arity and coefficient bounds. The problem is NP-hard for most choices of these parameters. As a response to this, we refine our understanding by analyzing the time complexity and the parameterized complexity (with respect to well-studied parameters such as primal and incidence treewidth). We obtain a comprehensive picture of the complexity landscape in both cases. Finally, we generalize our results to integer domains and sentences that are not in CNF.

This is an strongly extended version of two conference papers with the same authors that appeared at KR 2020 (Title: Fine-Grained Complexity of Temporal Problems) and AAAI 2021 (Title: Disjunctive Temporal Problems under Structural Restrictions)