Complexity Analyses of Bienstock–Zuckerberg and Lasserre Relaxations on the Matching and Stable Set Polytopes

Abstract

AbstractMany hierarchies of lift-and-project relaxations for 0,1 integer programs have been proposed, two of the most recent and strongest being those by Lasserre in 2001, and Bienstock and Zuckerberg in 2004. We prove that, on the LP relaxation of the matching polytope of the complete graph on (2n + 1) vertices defined by the nonnegativity and degree constraints, the Bienstock–Zuckerberg operator (even with positive semidefiniteness constraints) requires \(\Theta(\sqrt{n})\) rounds to reach the integral polytope, while the Lasserre operator requires Θ(n) rounds. We also prove that Bienstock–Zuckerberg operator, without the positive semidefiniteness constraint requires approximately n/2 rounds to reach the stable set polytope of the n-clique, if we start with the fractional stable set polytope. As a by-product of our work, we consider a significantly strengthened version of Sherali–Adams operator and a strengthened version of Bienstock–Zuckerberg operator. Most of our results also apply to these stronger operators.Keywordsmatching polytopelift-and-project methodsinteger programmingsemidefinite programmingconvex relaxations