Algebraic Theory of Promise Constraint Satisfaction Problems, First Steps

Abstract

What makes a computational problem easy (e.g., in P, that is, solvable in polynomial time) or hard (e.g., NP-hard)? This fundamental question now has a satisfactory answer for a quite broad class of computational problems, so called fixed-template constraint satisfaction problems (CSPs) -it has turned out that their complexity is captured by a certain specific form of symmetry. This paper explains an extension of this theory to a much broader class of computational problems, the promise CSPs, which includes relaxed versions of CSPs such as the problem of finding a 137coloring of a 3-colorable graph.

Boolean symmetric vs. functional PCSP dichotomy

“…Eqn 3 p ,1 → C + 3 p . That PCSP(1-in-3, C + k ) is tractable whenever k = 3 p or k = 2 × 3 p was first observed in [11]. If Problem 38 were answered in the affirmative then we would have that PCSP(1-in-3, C + k ) is tractable if and only if k = 3 p or k = 2 × 3 p .…”

“…For any k ∈ Z, define S k ⊆ Z a to be the set of vectors that sum up to k, with non-negative coordinates. 11 Lemma 30.…”

“…, t N , the last kN vectors in the sum above include at least k copies of each vector in R. By removing the k subtracted vectors from the k copies of each vector from R, we find that x can be written as a sum of k + 1 − k + kN = kN + 1 vectors from R, i.e. x ∈ (kN + 1)R. 11 We see the elements of Z a as frequency vectors, and the elements of (Z a ) r as tuples of frequency vectors.…”

1-in-3 vs. Not-All-Equal: Dichotomy of a broken promise

“…It was only resolved in certain special cases [KLS00, BG16, BBKO21, Hua13, KOWŽ23] or under strong complexity-theoretic assumptions such as variants of the Unique Games Conjecture [DMR09, GS20, BKLM21]. Other particular examples of PCSPs have been studied in [BBB21,Bra22,BŽ22,NŽ22]. The PCSP templates considered there have either small domains or specific structure.…”

“…The relational structures 1-in-3 and NAE contain a single, ternary relation, which is symmetric, i.e., it is invariant under permutations of the arguments, and it is rainbow-free, i.e., it does not contain any tuple (𝑥, 𝑦, 𝑧) whose arguments are all distinct (note that this is always the case for Boolean ternary structures). Following [BBB21], we describe this type of relational structures by associating digraphs to them. More precisely, given a digraph G = (𝑉 , 𝐸), we let G be the relational structure defined by G = (𝑉 ; {(𝑥, 𝑥, 𝑦), (𝑥, 𝑦, 𝑥), (𝑦, 𝑥, 𝑥) | (𝑥, 𝑦) ∈ 𝐸}).…”