Sherali-adams relaxations of the matching polytope

Claire Mathieu, Alistair Sinclair
2009 Proceedings of the 41st annual ACM symposium on Symposium on theory of computing - STOC '09  
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
more » ... + 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.
doi:10.1145/1536414.1536456 dblp:conf/stoc/MathieuS09 fatcat:4qkzz7y6tbghpjxtobd42jxss4