Abstract
We analyze how the standard reductions between constraint satisfaction problems affect their proof complexity. We show that, for the most studied propositional, algebraic, and semialgebraic proof systems, the classical constructions of pp-interpretability, homomorphic equivalence, and addition of constants to a core preserve the proof complexity of the CSP. As a result, for those proof systems, the classes of constraint languages for which small unsatisfiability certificates exist can be characterized algebraically. We illustrate our results by a gap theorem saying that a constraint language either has resolution refutations of constant width or does not have bounded-depth Frege refutations of subexponential size. The former holds exactly for the widely studied class of constraint languages of bounded width. This class is also known to coincide with the class of languages with refutations of sublinear degree in Sums of Squares and Polynomial Calculus over the real field, for which we provide alternative proofs. We then ask for the existence of a natural proof system with good behavior with respect to reductions and simultaneously small-size refutations beyond bounded width. We give an example of such a proof system by showing that bounded-degree Lovász-Schrijver satisfies both requirements. Finally, building on the known lower bounds, we demonstrate the applicability of the method of reducibilities and construct new explicit hard instances of the graph three-coloring problem for all studied proof systems.
Highlights
The notion of efficient reduction is at the heart of all subareas of computational complexity
Whether there is any connection between the two is an even less explored mystery. This fact is subject to revision, especially if proof complexity exports its methods to the study of problems beyond universal combinatorial statements
Ochremiak 110:3 known correspondance between local consistency algorithms, existential pebble games and bounded width resolution [17, 2], the lower bounds for propositional and semi-algebraic proof systems [1, 19, 6, 7, 13, 10], and a modest amount of additional work to fill in the gaps, we prove the following strong gap theorem: Theorem 1
Summary
The notion of efficient reduction is at the heart of all subareas of computational complexity. The goal of this paper is to develop the standard theory of reductions between constraint satisfaction problems in a way that it applies to many of the proof systems from the literature, including but not limited to SOS. Doing this requires a good amount of tedious work, but at the same time has some surprises to offer that we discuss next. It looks unlikely that our methods could be used for this proof system, whether it is closed under the standard CSP reducibilities is something that was not checked, neither in [3], nor here
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