An Algebraic Hardness Criterion for Surjective Constraint Satisfaction

TL;DR

This paper introduces an algebraic criterion based on polymorphism clones that determines the computational hardness of surjective CSPs on finite structures, establishing a new connection between algebraic properties and complexity.

Contribution

It provides the first algebraic condition that implies NP-hardness of surjective CSPs, linking algebraic structure to computational complexity.

Findings

Algebraic condition on polymorphism clone implies surjective CSP hardness

Surjective CSP is NP-complete for structures with only unary polymorphisms

First use of algebraic methods to infer surjective CSP complexity

Abstract

The constraint satisfaction problem (CSP) on a 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 condition on the polymorphism clone of B and prove that it is sufficient for the hardness of the surjective CSP on a finite structure B, in the sense that this problem admits a reduction from a certain fixed-structure CSP. To our knowledge, this is the first result that allows one to use algebraic information from a relational structure B to infer information on the complexity hardness of surjective constraint satisfaction on B. A corollary of our result is that, on any finite non-trivial structure having only essentially unary polymorphisms, surjective constraint satisfaction is NP-complete.