Lipschitz Continuous Algorithms for Covering Problems

TL;DR

This paper develops Lipschitz continuous algorithms for classical covering problems in graphs, ensuring stability against small input changes, and introduces a novel cycle sparsification technique for feedback vertex set approximation.

Contribution

It introduces the first Lipschitz continuous algorithms for covering problems and proposes a new cycle sparsification method for feedback vertex set.

Findings

Algorithms are stable under input perturbations.

Cycle sparsification improves feedback vertex set approximation.

Enhanced robustness of combinatorial algorithms.

Abstract

Combinatorial algorithms are widely used for decision-making and knowledge discovery, and it is important to ensure that their output remains stable even when subjected to small perturbations in the input. Failure to do so can lead to several problems, including costly decisions, reduced user trust, potential security concerns, and lack of replicability. Unfortunately, many fundamental combinatorial algorithms are vulnerable to small input perturbations. To address the impact of input perturbations on algorithms for weighted graph problems, Kumabe and Yoshida (FOCS'23) recently introduced the concept of Lipschitz continuity of algorithms. This work explores this approach and designs Lipschitz continuous algorithms for covering problems, such as the minimum vertex cover, set cover, and feedback vertex set problems. Our algorithm for the feedback vertex set problem is based on linear programming, and in the rounding process, we develop and use a technique called cycle sparsification, which may be of independent interest.