Mathematical Methods of Operations Research

, Volume 60, Issue 3, pp 349–367 | Cite as

On semidefinite programming relaxations for the satisfiability problem

  • Miguel F. AnjosEmail author


This paper is concerned with the analysis and comparison of semidefinite programming (SDP) relaxations for the satisfiability (SAT) problem. Our presentation is focussed on the special case of 3-SAT, but the ideas presented can in principle be extended to any instance of SAT specified by a set of boolean variables and a propositional formula in conjunctive normal form. We propose a new SDP relaxation for 3-SAT and prove some of its theoretical properties. We also show that, together with two SDP relaxations previously proposed in the literature, the new relaxation completes a trio of linearly sized relaxations with increasing rank-based guarantees for proving satisfiability. A comparison of the relative practical performances of the SDP relaxations shows that, among these three relaxations, the new relaxation provides the best tradeoff between theoretical strength and practical performance within an enumerative algorithm.


Satisfiability problem Semidefinite programming Combinatorial optimization Global optimization 


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The author thanks Franz Rendl, Monique Laurent, and Etienne de Klerk for several helpful discussions.

Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Operational Research Group, School of MathematicsUniversity of SouthamptonSouthamptonUnited Kingdom

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