The Power of the Combined Basic Linear Programming and Affine Relaxation for Promise Constraint Satisfaction Problems

Abstract

In the field of constraint satisfaction problems (CSPs), promise CSPs are an exciting new direction of study. In a promise CSP, each constraint comes in two forms: ``strict"" and ``weak,"" and in the associated decision problem one must distinguish between being able to satisfy all the strict constraints versus not being able to satisfy all the weak constraints. The most commonly cited example of a promise CSP is the approximate graph coloring problem---which has recently seen exciting progress [Bul\' {\i} n, Krokhin, and Opr\u sal, Proceedings of the Symposium on Theory of Computing, 2019, pp. 602--613 and Wrochna and \v Zivn\' y, Proceedings of the Symposium on Discrete Algorithms, 2020, pp. 1426--1435] benefiting from a systematic algebraic approach to promise CSPs based on ``polymorphisms,"" operations that map tuples in the strict form of each constraint to tuples in the corresponding weak form. In this work, we present a simple algorithm which in polynomial time solves the decision problem for all promise CSPs that admit infinitely many symmetric polymorphisms, which are invariant under arbitrary coordinate permutations. This generalizes previous work of the first two authors [Brakensiek and Guruswami, Proceedings of the Symposium on Discrete Algorithms, 2019, pp. 436--455]. We also extend this algorithm to a more general class of block-symmetric polymorphisms. As a corollary, this single algorithm solves all polynomial-time tractable Boolean CSPs simultaneously. These results give a new perspective on Schaefer's classic dichotomy theorem and shed further light on how symmetries of polymorphisms enable algorithms. Finally, we show that block symmetric polymorphisms are not only sufficient but also necessary for this algorithm to work, thus establishing its precise power.

Algebraic Approach to Promise Constraint Satisfaction

“…It would be also very interesting to develop a general theory of how such a structure D can be constructed from bipartite minor The polynomial-time algorithms of Bulatov and Zhuk [36,92,93] that solve all tractable finitedomain CSPs are quite involved and both use deep structural analysis of finite algebras. Can algorithms for tractable PCSPs that involve infinite-domain CSPs (e.g., such as those in References [30,31]) lead to a new simpler algorithm that solves all tractable CSPs?…”

Beyond PCSP (1-in-3,NAE)

“…In the proof of one of our results (Theorem 10) we will use Theorem 9 to rule out solvability by BLP+AIP, the currently strongest known algorithm for PCSPs [12]. However, for our tractability results, we will only need a characterisation result (in terms of polymorphisms) of a weaker relaxation, namely the affine IP relaxation (AIP) [5], which also works for the search version of PCSPs; details are in Section 4.…”

“…In all other cases, the resulting PCSP is not solved by the combined basic LP and affine IP relaxation of Brakensiek and Guruswami [11], the currently strongest known algorithm for PCSPs. The power of this relaxation has recently been characterised by Brakensiek, Guruswami, Wrochna, and Živný [12].…”

CLAP: A New Algorithm for Promise CSPs

“…The BLP+AIP algorithm does that. A follow-up work [23] established the power of BLP+AIP in terms of a minion and (a property of) polymorphisms. The minion capturing BLP+AIP is essentially a product of the BLP and AIP minions [23].…”

“…A follow-up work [23] established the power of BLP+AIP in terms of a minion and (a property of) polymorphisms. The minion capturing BLP+AIP is essentially a product of the BLP and AIP minions [23]. Concerning polymorphisms, BLP+AIP is captured by polymorphisms of all odd arities that are invariant under permutations that only permute odd and even coordinates.…”