An $O(1)$-Approximation Algorithm for Dynamic Weighted Vertex Cover with Soft Capacity

TL;DR

This paper presents a deterministic primal-dual algorithm that maintains a constant-factor approximate solution for the dynamic capacitated vertex cover problem with efficient update times, extending to more general models.

Contribution

It extends previous work by providing an $O(1)$-approximation algorithm with $O(rac{ ext{log} n}{ ext{epsilon}})$ amortized update time for the dynamic capacitated vertex cover problem.

Findings

Achieves a constant-factor approximation in dynamic settings.

Provides an efficient $O(rac{ ext{log} n}{ ext{epsilon}})$ amortized update time.

Extends to nonuniform demands and capacitated set cover problems.

Abstract

This study considers the (soft) capacitated vertex cover problem in a dynamic setting. This problem generalizes the dynamic model of the vertex cover problem, which has been intensively studied in recent years. Given a dynamically changing vertex-weighted graph $G=(V,E)$, which allows edge insertions and edge deletions, the goal is to design a data structure that maintains an approximate minimum vertex cover while satisfying the capacity constraint of each vertex. That is, when picking a copy of a vertex $v$ in the cover, the number of $v$'s incident edges covered by the copy is up to a given capacity of $v$. We extend Bhattacharya et al.'s work [SODA'15 and ICALP'15] to obtain a deterministic primal-dual algorithm for maintaining a constant-factor approximate minimum capacitated vertex cover with $O(\log n / \epsilon)$ amortized update time, where $n$ is the number of vertices in the graph. The algorithm can be extended to (1) a more general model in which each edge is associated with a nonuniform and unsplittable demand, and (2) the more general capacitated set cover problem.