Bilu-linial stable instances of max cut and minimum multiway cut

Abstract

We investigate the notion of stability proposed by Bilu and Linial. We obtain an exact polynomial-time algorithm for γ-stable Max Cut instances with γ ≥ c√log n log log n for some absolute constant c > 0. Our algorithm is robust: it never returns an incorrect answer; if the instance is γ-stable, it finds the maximum cut, otherwise, it either finds the maximum cut or certifies that the instance is not γ-stable. We prove that there is no robust polynomial-time algorithm for γ-stable instances of Max Cut when γ 0) [Bilu, Daniely, Linial, and Saks]. No hardness results were known for the problem.Additionally, we present an exact robust polynomial-time algorithm for 4-stable instances of Minimum Multiway Cut.We also study a relaxed notion of weak stability and present algorithms for weakly stable instances of Max Cut and Minimum Multiway Cut.