# Algorithms and Complexity of Difference Logic

arXiv: https://arxiv.org/abs/2402.03273 · HTML: https://mlsanity.com/paper/2402.03273/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Feb 2024.

## Built on by (most cited first, 1 of 1)

- [Maximum Satisfiability of Simple Temporal Problems](https://mlsanity.com/paper/2607.23785/) (2026)

## Abstract (arXiv)

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.