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 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 < αSC(n/2), where αSC is the best approximation factor for Sparsest Cut with non-uniform demands. That suggests that solving γ-stable instances with might be difficult or even impossible. Our algorithm is based on semidefinite programming. We show that the standard SDP relaxation for Max Cut (with triangle inequalities) is integral if , where is the least distortion with which every n point metric space of negative type embeds into ℓ1. On the negative side, we show that the SDP relaxation is not integral when . Moreover, there is no tractable convex relaxation for γ-stable instances of Max Cut when γ < αSC(n/2). Our results significantly improve previously known results. The best previously known algorithm for γ-stable instances of Max Cut required that (for some c > 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.