, Volume 46, Issue 1, pp 81-102

A Framework for Certified Boolean Branch-and-Bound Optimization

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Abstract

We consider optimization problems of the form (S, cost), where S is a clause set over Boolean variables x 1 ... x n , with an arbitrary cost function \(\mathit{cost}\colon \mathbb{B}^n \rightarrow \mathbb{R}\) , and the aim is to find a model A of S such that cost(A) is minimized. Here we study the generation of proofs of optimality in the context of branch-and-bound procedures for such problems. For this purpose we introduce \(\mathtt{DPLL_{BB}}\) , an abstract DPLL-based branch-and-bound algorithm that can model optimization concepts such as cost-based propagation and cost-based backjumping. Most, if not all, SAT-related optimization problems are in the scope of \(\mathtt{DPLL_{BB}}\) . Since many of the existing approaches for solving these problems can be seen as instances, \(\mathtt{DPLL_{BB}}\) allows one to formally reason about them in a simple way and exploit the enhancements of \(\mathtt{DPLL_{BB}}\) given here, in particular its uniform method for generating independently verifiable optimality proofs.

Partially supported by Spanish Min. of Science and Innovation through the projects TIN2009-13591-C02-01 and TIN2007-68093-C02-01 (LogicTools-2).