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 $γ\geq c\sqrt{\log n}\log\log n$ 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 $γ< α_{SC}(n/2)$, 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 $\ell_2^2$ triangle inequalities) is integral if $γ\geq D_{\ell_2^2\to \ell_1}(n)$, where $D_{\ell_2^2\to \ell_1}(n)$ is the least distortion with which every $n$ point metric space of negative type embeds into $\ell_1$. On the negative side, we show that the SDP relaxation is not integral when $γ< D_{\ell_2^2\to \ell_1}(n/2)$. 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 $γ=o(\sqrt{\log n})$ 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 $γ\geq c\sqrt{n}$ (for some $c > 0$) [Bilu, Daniely, Linial, and Saks]. No hardness results were known for the problem. Additionally, we present an algorithm for 4-stable instances of Minimum Multiway Cut. We also study a relaxed notion of weak stability.

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