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

Lucy Ham; Marcel Jackson

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arXiv · 1611.00886

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

Lucy Ham·Marcel Jackson

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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.

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Lucy Ham, Marcel Jackson. 2016-11-03. All or nothing: toward a promise problem dichotomy for constraint problems. https://arxiv.org/abs/1611.00886

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