Affiliation:
1. Universitat Politècnica de Catalunya, Barcelona, Catalonia, Spain
2. Université Paris Diderot, Paris, France
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.
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Mathematics,Logic,General Computer Science,Theoretical Computer Science
Reference49 articles.
1. The complexity of the pigeonhole principle
2. Space Complexity in Propositional Calculus
3. Lower bounds for polynomial calculus: Non-binomial case;Alekhnovich M.;Proc. Steklov Institute Math.,2003
Cited by
9 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献
1. On vanishing sums of roots of unity in polynomial calculus and sum-of-squares;computational complexity;2023-11-12
2. Graph Colouring Is Hard on Average for Polynomial Calculus and Nullstellensatz;2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS);2023-11-06
3. On the strength of Sherali-Adams and Nullstellensatz as propositional proof systems;Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science;2022-08-02
4. On the complexity of CSP-based ideal membership problems;Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing;2022-06-09
5. ℋ-Colouring Dichotomy in Proof Complexity;Journal of Logic and Computation;2021-04-24