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 polynomialtime 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. Our algorithm is based on semidefinite programming. We show that the standard SDP relaxation for Max Cut (with ℓ 2 2 triangle inequalities) is integral if γ ≥ D ℓ 2 2 →ℓ1 (n), where D ℓ 2 2 →ℓ1 (n) is the least distortion with which every n point metric space of negative type embeds into ℓ 1 . On the negative side, we show that the SDP relaxation is not integral when γ < D ℓ 2 2 →ℓ1 (n/2). Moreover, there is no tractable convex relaxation for γ-stable instances of Max Cut when γ < α SC (n/2). That suggests that solving γ-stable instances with γ = o( √ log n) might be difficult or impossible. 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.
Later Makarychev et al. REF gave an algorithm which required only (c log n log log n)-multiplicative perturbation resilience for some constant c > 0 to retrieve the optimal Max-Cut in polynomial time.
Bilu-Linial Stable Instances of Max Cut and Minimum Multiway Cut
arXiv: Data Structures and Algorithms
Computer Science Mathematics