LP-Based Robust Algorithms for Noisy Minor-Free and Bounded Treewidth Graphs

TL;DR

This paper introduces a linear programming-based framework for solving optimization problems on noisy minor-free and bounded treewidth graphs, achieving near-optimal approximations despite adversarial corruptions.

Contribution

It provides the first general LP-based approach for noisy optimization problems on these graph classes, extending beyond independent set to MAX-k-CSPs.

Findings

Achieves a (1 + O(\delta \, log m \, log \, log m))-approximation for noisy MAX-k-CSPs.

Extends noisy optimization techniques to minor-free and bounded treewidth graphs.

Offers a unified LP-based framework applicable to various problems in noisy graph settings.

Abstract

We give a general approach for solving optimization problems on noisy minor free graphs, where a \delta-fraction of edges and vertices are adversarially corrupted. The noisy setting was first considered by Magen and Moharrami and they gave a (1 + \delta)-estimation algorithm for the independent set problem. Later, Chan and Har-Peled designed a local search algorithm that finds a (1 + O(\delta))-approximate independent set. However, nothing was known regarding other problems in the noisy setting. Our main contribution is a general LP-based framework that yields a (1 + O(\delta log m log log m))-approximation algorithm for noisy MAX-k-CSPs on m clauses.