ℋ-Colouring Dichotomy in Proof Complexity

Abstract

Abstract The $\mathcal {H}$-colouring problem for undirected simple graphs is a computational problem from a huge class of the constraint satisfaction problems (CSPs): an $\mathcal {H}$-colouring of a graph $\mathcal {G}$ is just a homomorphism from $\mathcal {G}$ to $\mathcal {H}$ and the problem is to decide for fixed $\mathcal {H}$, given $\mathcal {G}$, if a homomorphism exists or not. The dichotomy theorem for the $\mathcal {H}$-colouring problem was proved by Hell and Nešetřil (1990, J. Comb. Theory Ser. B, 48, 92–110) (an analogous theorem for all CSPs was recently proved by Zhuk (2020, J. ACM, 67, 1–78) and Bulatov (2017, FOCS, 58, 319–330)), and it says that for each $\mathcal {H}$, the problem is either $p$-time decidable or $NP$-complete. Since negations of unsatisfiable instances of CSP can be expressed as propositional tautologies, it seems to be natural to investigate the proof complexity of CSP. We show that the decision algorithm in the $p$-time case of the $\mathcal {H}$-colouring problem can be formalized in a relatively weak theory and that the tautologies expressing the negative instances for such $\mathcal {H}$ have polynomial proofs in propositional proof system $R^*(log)$, a mild extension of resolution. In fact, when the formulas are expressed as unsatisfiable sets of clauses, they have $p$-size resolution proofs. To establish this, we use a well-known connection between theories of bounded arithmetic and propositional proof systems. This upper bound follows also from a different construction in [1]. We complement this result by a lower bound result that holds for many weak proof systems for a special example of $NP$-complete case of the $\mathcal {H}$-colouring problem, using known results about the proof complexity of the pigeonhole principle. The main goal of our work is to start the development of some of the theories beyond the CSP dichotomy theorem in bounded arithmetic. We aim eventually—in a subsequent work—to formalize in such a theory the soundness of Zhuk’s algorithm, extending the upper bound proved here from undirected simple graphs to the general case of directed graphs in some logical calculi.