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

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 7-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 < α_sc(n/2), where α_sc is the best approximation factor for Sparsest Cut with non-uniform demands. That suggests that solving γ-stable instances with γ = o( √log n) might be difficult or even impossible. Our algorithm is based on semidefinite programming. We show that the standard SDP relaxation for Max Cut (with ℓ²₂ triangle inequalities) is integral if γ > D_l22→ℓ₁(n), where D₁²2 →ℓ₁n) 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 γ < D _ℓ²→ℓ₁(n/2). Moreover, there is no tractable convex relaxation for γ-stable instances of Max Cut when γ < α_sc(n/2)- Our results significantly improve previously known results. The best previously known algorithm for γ- stable instances of Max Cut required that γ ≥ c√n(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.