Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lipschitz Continuous Algorithms for Covering Problems

Soh Kumabe, Yuichi Yoshida

TL;DR

This work designs Lipschitz continuous algorithms for covering problems, such as the minimum vertex cover, set cover, and feedback vertex set problems, and develops and uses a technique called cycle sparsification, which may be of independent interest.

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.

Lipschitz Continuous Algorithms for Covering Problems

TL;DR

This work designs Lipschitz continuous algorithms for covering problems, such as the minimum vertex cover, set cover, and feedback vertex set problems, and develops and uses a technique called cycle sparsification, which may be of independent interest.

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.
Paper Structure (44 sections, 41 theorems, 131 equations, 1 table, 11 algorithms)

This paper contains 44 sections, 41 theorems, 131 equations, 1 table, 11 algorithms.

Table of Contents

  1. Introduction
  2. Lipschitz Continuity
  3. Our Results
  4. Proof Overview
  5. Vertex Cover
  6. Naive algorithm for Set Cover
  7. Greedy-based algorithm for Set Cover
  8. LP-based algorithm for Set Cover
  9. Feedback Vertex Set
  10. Lipschitz Continuity under Shared Randomness
  11. Related Work
  12. Preliminaries
  13. Organization
  14. Vertex Cover
  15. Approximation Ratio
  16. ...and 29 more sections

Key Result

Lemma 1.3

Let $\mathcal{A}$ be an algorithm that takes a graph $G=(V,E)$ and a weight vector $w \in \mathbb{R}_{\geq 0}^V$ and outputs a subset of $V$. Suppose that there exist some $c>0$ and $L>0$ such that holds for any $v\in V$, $w\in \mathbb{R}_{\geq 0}^V$, and $\delta > 0$ with either $\delta\leq c\cdot w_v$ or $w_v=0$. Then, $\mathcal{A}$ has Lipschitz constant $L$ on $G$. Similarly, if the above ine

Theorems & Definitions (75)

  • Definition 1.1: Lipschitz constant of a randomized algorithm kumabe2023lipschitz
  • Definition 1.2: Lipschitz constant of a randomized algorithm with shared randomness kumabe2023lipschitz
  • Lemma 1.3
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 65 more