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.
This result was improved by Markarychev et al. REF , who showed the standard SDP relaxation is integral for Ω( √ log n log log n)-perturbation resilient instances.
Bilu-Linial Stable Instances of Max Cut and Minimum Multiway Cut
arXiv: Data Structures and Algorithms
Computer Science Mathematics