Algebraic Global Gadgetry for Surjective Constraint Satisfaction

TL;DR

This paper introduces an algebraic framework for establishing hardness results in surjective constraint satisfaction problems, unifying various intractability proofs and simplifying reductions from classical CSPs.

Contribution

The authors develop a novel algebraic framework that computes global gadgetry to reduce classical CSPs to surjective CSPs, enabling unified hardness proofs.

Findings

Proves hardness of the disconnected cut problem

Establishes intractability of no-rainbow 3-coloring

Shows surjective CSP intractability on all 2-element structures

Abstract

The constraint satisfaction problem (CSP) on a finite relational structure B is to decide, given a set of constraints on variables where the relations come from B, whether or not there is a assignment to the variables satisfying all of the constraints; the surjective CSP is the variant where one decides the existence of a surjective satisfying assignment onto the universe of B. We present an algebraic framework for proving hardness results on surjective CSPs; essentially, this framework computes global gadgetry that permits one to present a reduction from a classical CSP to a surjective CSP. We show how to derive a number of hardness results for surjective CSP in this framework, including the hardness of the disconnected cut problem, of the no-rainbow 3-coloring problem, and of the surjective CSP on all 2-element structures known to be intractable (in this setting). Our framework thus allows us to unify these hardness results, and reveal common structure among them; we believe that our hardness proof for the disconnected cut problem is more succinct than the original. In our view, the framework also makes very transparent a way in which classical CSPs can be reduced to surjective CSPs.