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 ⊆ 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 − ) against an adaptive adversary and of α GW ≈ 0.8786 against an oblivious adversary (here α GW is the approximation achieved by the random hyperplane algorithm of [Goemans-Williamson J. ACM '95]). Additionally, we present a hardness of approximation of α 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 multiobjective optimization to LP duality and the ellipsoid algorithm to obtain our results.