On the Width of Semi-Algebraic Proofs and Algorithms

In this paper we initiate the study of width in semi-algebraic proof systems and various cut-based procedures in integer programming. We focus on two important systems: Gomory-Chvatal cutting planes and Lovasz-Schrijver lift-and-project procedures. We develop general methods for proving width lower bounds and apply them to random k-CNFs and several popular combinatorial principles like the perfect matching principle and Tseitin tautologies. We also show how to apply our methods to various combinatorial optimization problems. We establish an “ultimate” trade-off between width and rank, that is give an example in which small width proofs are possible but require exponentially many rounds to perform them.

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