An Algorithmic Blend of LPs and Ring Equations for Promise CSPs

Abstract

Promise CSPs are a relaxation of constraint satisfaction problems where the goal is to find an assignment satisfying a relaxed version of the constraints. Several well known problems can be cast as promise CSPs including approximate graph and hypergraph coloring, discrepancy minimization, and interesting variants of satisfiability. Similar to CSPs, the tractability of promise CSPs can be tied to the structure of associated operations on the solution space called (weak) polymorphisms. However, compared to CSPs whose polymorphisms are well-structured algebraic objects called clones, polymorphisms in the promise world are much less constrained -essentially any infinite family of functions obeying mild conditions can arise as polymorphisms. Under the thesis that non-trivial polymorphisms govern tractability, promise CSPs therefore provide a fertile ground for the discovery of novel algorithms.

Promises Make Finite (Constraint Satisfaction) Problems Infinitary

“…Fortunately, building on the initial insights and results in [1,9,11,10], it was observed in [14] (among many other important results such as the NPhardness of 5-coloring a 3-colorable graph) that the basics of the CSP theory from [4] generalize to PCSPs. In particular, the notions of pp-constructions and polymorphisms have their PCSP counterparts and the connection between relational and algebraic structures works just as well as in the CSP.…”

“…A fundamental question is whether each finite tractable PCSP template can be pp-constructed from an infinite tractable CSP template. In [10], the authors conjectured that the answer is positive and even suggested a family of tractable CSPs that might solve all Boolean PCSPs.…”

“…The complexity of the former problem is a notorious open question (for a recent development see [9,14], but even 6-coloring a 3-colorable graph is not known to be NP-complete). On the other hand, the latter problem admits an elegant polynomial time algorithm [11,10], which we now describe.…”

Algebraic Theory of Promise Constraint Satisfaction Problems, First Steps

“…This problem, even its search version, admits elegant polynomial time algorithms [14,15] -one is based on solving linear equations over the integers, another one on linear programming. For this specific template, the algorithm can be further simplified as follows.…”

“…After the introduction of the PCSP framework, it has quickly turned out that both the notion of a polymorphism and Theorem 2.10 have straightforward generalizations. Any PCSP whose template (over the domains E 2 and E 2 ) admits all these polymorphisms is in P [14,15]. sketch.…”