All or nothing: toward a promise problem dichotomy for constraint problems

TL;DR

This paper explores a dichotomy in constraint satisfaction problems, showing that intractability can be characterized by an infinite hierarchy of promise problems related to partial solutions, extending classical NP-hardness results.

Contribution

It introduces a hierarchy of promise problems that characterize intractability of CSPs, generalizing known NP-hardness results to a broader class of problems.

Findings

Intractability of CSPs can be replaced by an infinite hierarchy of promise problems.

For any fixed k, distinguishing certain graph colorability cases is computationally hard.

The main result applies to all known intractable constraint problems over finite languages.

Abstract

A finite constraint language $\mathscr{R}$ is a finite set of relations over some finite domain $A$. We show that intractability of the constraint satisfaction problem $\operatorname{CSP}(\mathscr{R})$ can, in all known cases, be replaced by an infinite hierarchy of intractable promise problems of increasingly disparate promise conditions: where instances are guaranteed to either have no solutions at all, or to be $k$-robustly satisfiable (for any fixed $k$), meaning that every "reasonable" partial instantiation on~$k$ variables extends to a solution. For example, subject to the assumption $\texttt{P}\neq \texttt{NP}$, then for any~$k$, we show that there is no polynomial time algorithm that can distinguish non-$3$-colourable graphs, from those for which any reasonable $3$-colouring of any $k$ of the vertices can extend to a full $3$-colouring. Our main result shows that an analogous statement holds for all known intractable constraint problems over fixed finite constraint languages.