Fault Tolerant Max-Cut

TL;DR

This paper studies fault tolerant Max-Cut, proposing approximation algorithms that are nearly optimal against different adversaries, and establishing hardness results that show the limits of approximation.

Contribution

It introduces the first approximation algorithms for fault tolerant Max-Cut with adversarial vertex removal and proves tight hardness bounds.

Findings

Achieves a (0.878 - ε)-approximation against adaptive adversaries.

Achieves approximately 0.8786 approximation against oblivious adversaries.

Proves hardness of approximation matching the Goemans-Williamson ratio.

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.