Combinatorial Gap Theorem and Reductions between Promise CSPs

TL;DR

This paper introduces a combinatorial gap theorem for CSPs, providing a new perspective on the hardness of approximation and establishing simple reductions between Promise CSPs, with implications for NP-hardness proofs.

Contribution

It presents a novel combinatorial gap theorem for CSPs and uses it to derive simple, PCP-free NP-hardness proofs and reductions between Promise CSPs.

Findings

A new gap theorem bounds the value of unsolvable instances away from one.

The gap theorem implies NP-hardness of certain approximation problems.

Every CSP can be reduced to known NP-hard Promise CSPs.

Abstract

A value of a CSP instance is typically defined as a fraction of constraints that can be simultaneously met. We propose an alternative definition of a value of an instance and show that, for purely combinatorial reasons, a value of an unsolvable instance is bounded away from one; we call this fact a gap theorem. We show that the gap theorem implies NP-hardness of a gap version of the Layered Label Cover Problem. The same result can be derived from the PCP Theorem, but a full, self-contained proof of our reduction is quite short and the result can still provide PCP-free NP-hardness proofs for numerous problems. The simplicity of our reasoning also suggests that weaker versions of Unique-Games-type conjectures, e.g., the d-to-1 conjecture, might be accessible and serve as an intermediate step for proving these conjectures in their full strength. As the second, main application we provide a sufficient condition under which a fixed template Promise Constraint Satisfaction Problem (PCSP) reduces to another PCSP. The correctness of the reduction hinges on the gap theorem, but the reduction itself is very simple. As a consequence, we obtain that every CSP can be canonically reduced to most of the known NP-hard PCSPs, such as the approximate hypergraph coloring problem.