The Power of the Combined Basic LP and Affine Relaxation for Promise CSPs

Abstract

In the field of constraint satisfaction problems (CSP), 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 [BKO19, WZ20] 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 [BG19]. 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

17 pages, to appear in SICOMP

Cited by in corpus (12)

• Algebraic approach to promise constraint satisfaction
• CLAP: A New Algorithm for Promise CSPs
• Conditional Dichotomy of Boolean Ordered Promise CSPs
• Local consistency as a reduction between constraint satisfaction problems
• An invitation to the promise constraint satisfaction problem
• Beyond PCSP (1-in-3,NAE)
• The combined basic LP and affine IP relaxation for promise VCSPs on infinite domains
• On the complexity of symmetric vs. functional PCSPs
• Topology and adjunction in promise constraint satisfaction
• Approximate Graph Colouring and the Crystal with a Hollow Shadow
• Hierarchies of Minion Tests for PCSPs through Tensors
• Solving promise equations over monoids and groups