Limitations of Affine Integer Relaxations for Solving Constraint Satisfaction Problems

DROPS (Schloss Dagstuhl – Leibniz Center for Informatics)

Abstract

We show that various recent algorithms for finite-domain constraint satisfaction problems (CSP), which are based on solving their affine integer relaxations, do not solve all tractable and not even all Maltsev CSPs. This rules them out as candidates for a universal polynomial-time CSP algorithm. The algorithms are ℤ-affine k-consistency, BLP+AIP, BA^{k}, and CLAP. We thereby answer a question by Brakensiek, Guruswami, Wrochna, and Živný [Joshua Brakensiek et al., 2020] whether a constant level of BA^{k}solves all tractable CSPs in the negative: Indeed, not even a sublinear level k suffices. We also refute a conjecture by Dalmau and Opršal [Víctor Dalmau and Jakub Opršal, 2024] (LICS 2024) that every CSP is either solved by ℤ-affine k-consistency or admits a Datalog reduction from 3-colorability. For the cohomological k-consistency algorithm, that is also based on affine relaxations, we show that it correctly solves our counterexample but fails on an NP-complete template.

Authors 1

1. Benedikt Pago

   University of Cambridge

   Affiliation as printed

   University of Cambridge, UK