Modular Counting CSP: Reductions and Algorithms
Amirhossein Kazeminia, Andrei A.Bulatov

TL;DR
This paper advances the understanding of counting solutions modulo an integer in the general CSP, providing algorithms for specific cases and hardness results, extending previous graph-based classifications to broader structures.
Contribution
It introduces new complexity classifications and algorithms for counting solutions modulo p in general CSPs, highlighting structural differences from graph-based cases.
Findings
Algorithm for p=2 under certain conditions
Hardness results based on automorphism properties
Reduction to binary relations with strong conditions
Abstract
The Constraint Satisfaction Problem (CSP) is ubiquitous in various areas of mathematics and computer science. Many of its variations have been studied including the Counting CSP, where the goal is to find the number of solutions to a CSP instance. The complexity of finding the exact number of solutions of a CSP is well understood (Bulatov, 2013, and Dyer and Richerby, 2013) and the focus has shifted to other variations of the Counting CSP such as counting the number of solutions modulo an integer. This problem has attracted considerable attention recently. In the case of CSPs based on undirected graphs Bulatov and Kazeminia (STOC 2022) obtained a complexity classification for the problem of counting solutions modulo p for arbitrary prime p. In this paper we report on the progress made towards a similar classification for the general CSP, not necessarily based on graphs. We identify…
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