Certified Symmetry and Dominance Breaking for Combinatorial Optimisation

Abstract

Symmetry and dominance breaking can be crucial for solving hard combinatorial search and optimisation problems, but the correctness of these techniques sometimes relies on subtle arguments. For this reason, it is desirable to produce efficient, machine-verifiable certificates that solutions have been computed correctly. Building on the cutting planes proof system, we develop a certification method for optimisation problems in which symmetry and dominance breaking are easily expressible. Our experimental evaluation demonstrates that we can efficiently verify fully general symmetry breaking in Boolean satisfiability (SAT) solving, thus providing, for the first time, a unified method to certify a range of advanced SAT techniques that also includes XOR and cardinality reasoning. In addition, we apply our method to maximum clique solving and constraint programming as a proof of concept that the approach applies to a wider range of combinatorial problems.

QMaxSATpb: A Certified MaxSAT Solver

“…If the solver is replaced by any other modern SAT solver with proof logging capabilities, only minor syntactic modifications are needed to make it VeriPB-compatible. Indeed, as mentioned before, redundance-based strengthening generalizes the well-known RAT rule, and moreover, VeriPB can additionally handle symmetry breaking, cardinality reasoning and XOR reasoning [25,7].…”

“…We now review the rules of the VeriPB proof system we use for certification of QMaxSAT; we refer the reader to Bogaerts et al [7] for an exposition of the full proof system. A pseudo-Boolean (PB) constraint C is a linear inequality of the form i a i l i ≥ A where a i and A are integers, l i are literals.…”

“…One promising proof format to fill this gap is VeriPB [17,25,24,22], a proof format for pseudo-Boolean satisfiability, that was recently extended to pseudo-Boolean optimization [7], which generalizes MaxSAT. VeriPB builds on top of the cutting planes proof system [12], and extends it with a generalization of the Resolution Asymmetric Tautology (RAT) rule that lies at the basis of the most successful proof formats for SAT.…”

“…VeriPB builds on top of the cutting planes proof system [12], and extends it with a generalization of the Resolution Asymmetric Tautology (RAT) rule that lies at the basis of the most successful proof formats for SAT. VeriPB naturally facilitates proof logging for advanced techniques such as XOR and cardinality reasoning [25] and symmetry breaking [7] that have been largely out of reach of other proof formats for SAT.…”

Certifying Parity Reasoning Efficiently Using Pseudo-Boolean Proofs

“…The last couple of years have witnessed quite significant developments in proof logging. Since the conference version of this paper appeared, our pseudo-Boolean proof logging method has been extended further to deal with fully general symmetry breaking in SAT solving [BGMN22], and also to support pseudo-Boolean solving using SAT solvers [GMNO22]. Furthermore, there have been promising preliminary results on providing proof logging for MaxSAT solvers [VWB22] and constraint programming solvers [GMN22].…”