Sherali-adams relaxations of the matching polytope

Authors:
Claire Mathieu;Alistair Sinclair
Affiliations:
Brown University, Providence, RI, USA;University of California, Berkeley, CA, USA
Venue:
Proceedings of the forty-first annual ACM symposium on Theory of computing
Year:
2009

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Abstract

We study the Sherali-Adams lift-and-project hierarchy of linear programming relaxations of the matching polytope. Our main result is an asymptotically tight expression 1+1/k for the integrality gap after k rounds of this hierarchy. The result is derived by a detailed analysis of the LP after k rounds applied to the complete graph K_{2d+1}. We give an explicit recurrence for the value of this LP, and hence show that its gap exhibits a "phase transition," dropping from close to its maximum value 1+1/2d to close to 1 around the threshold k=2d-Θ(√d). We also show that the rank of the matching polytope (i.e., the number of Sherali-Adams rounds until the integer polytope is reached) is exactly 2d-1.