Algorithms and Complexity of Difference Logic
<|reference_start|>Algorithms and Complexity of Difference Logic: 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.<|reference_end|>
@article{dabrowski2024algorithms,
    title={Algorithms and Complexity of Difference Logic},
    author={Konrad K. Dabrowski, Peter Jonsson, Sebastian Ordyniak, George Osipov},
    journal={arXiv preprint arXiv:2402.03273},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.03273},
    primaryClass={cs.DS cs.LO}
}