Complete Randomized Cutting Plane Algorithms for Propositional Satisfiability

The propositional satisfiability problem (SAT) is a fundamental problem in computer science and combinatorial optimization. A considerable number of prior researchers have investigated SAT, and much is already known concerning limitations of known algorithms for SAT. In particular, some necessary conditions are known, such that any algorithm not meeting those conditions cannot be efficient. This paper reports a research to develop and test a new algorithm that meets the currently known necessary conditions. In chapter three, we give a new characterization of the convex integer hull of SAT, and two new algorithms for finding strong cutting planes. We also show the importance of choosing which vertex to cut, and present heuristics to find a vertex that allows a strong cutting plane. In chapter four, we describe an experiment to implement a SAT solving algorithm using the new algorithms and heuristics, and to examine their effectiveness on a set of problems. In chapter five, we describe the implementation of the algorithms, and present computational results. For an input SAT problem, the output of the implemented program provides either a witness to the satisfiability or a complete cutting plane proof of satisfiability. The description, implementation, and testing of these algorithms yields both empirical data to characterize the performance of the new algorithms, and additional insight to further advance the theory. We conclude from the computational study that cutting plane algorithms are efficient for the solution of a large class of SAT problems

Cutting plane and Frege proofs

The cutting plane refutation system CP for propositional logic is an extension of resolution and is based on showing the non-existence of solutions for families of integer linear inequalities. We define the system CP + , a modification of the cutting plane system, and show that CP + can polynomially simulate Frege systems F . In [8], it is shown that F polynomially simulates CP + , so both systems are polynomially equivalent. To establish this result, propositional formulas are represented in a natural manner, and an effective version of cut elimination is proved for the system CP + . Additionally, an alternative proof is given which directly translates F proofs into CP + . Thus, within a polynomial factor, one can simulate classical propositional logic proofs using the cut rule by refutation-style proofs involving linear inequalities with the ceiling operation. Since there are polynomial size cutting plane CP proofs for many elementary combinatorial principles (pigeonhole p..