Fault Tolerant Max-Cut

Abstract

In this work, we initiate the study of fault tolerant Max Cut, where given an edge-weighted undirected graph $G=(V,E)$, the goal is to find a cut $S\subseteq V$ that maximizes the total weight of edges that cross $S$ even after an adversary removes $k$ vertices from $G$. We consider two types of adversaries: an adaptive adversary that sees the outcome of the random coin tosses used by the algorithm, and an oblivious adversary that does not. For any constant number of failures $k$ we present an approximation of $(0.878-\epsilon)$ against an adaptive adversary and of $\alpha_{GW}\approx 0.8786$ against an oblivious adversary (here $\alpha_{GW}$ is the approximation achieved by the random hyperplane algorithm of [Goemans-Williamson J. ACM `95]). Additionally, we present a hardness of approximation of $\alpha_{GW}$ against both types of adversaries, rendering our results (virtually) tight. The non-linear nature of the fault tolerant objective makes the design and analysis of algorithms harder when compared to the classic Max Cut. Hence, we employ approaches ranging from multi-objective optimization to LP duality and the ellipsoid algorithm to obtain our results.