Fully Dynamic Set Cover via Hypergraph Maximal Matching: An Optimal Approximation Through a Local Approach

TL;DR

This paper introduces an optimal, fully dynamic set cover algorithm with an approximation ratio matching the maximum set membership of elements, using a novel hypergraph maximal matching approach with efficient update times.

Contribution

It presents the first dynamic set cover algorithm achieving an approximation ratio exactly equal to the maximum set membership, with update time independent of universe size, via a new hypergraph maximal matching method.

Findings

Achieves $O(f^2)$ update time for dynamic set cover with $f$-approximation.

Provides a hypergraph maximal matching algorithm with $O(r^2)$ expected update time.

Proves the approximation ratio is optimal under the unique games conjecture.

Abstract

In the (fully) dynamic set cover problem, we have a collection of $m$ sets from a universe of size $n$ that undergo element insertions and deletions; the goal is to maintain an approximate set cover of the universe after each update. We give an $O(f^2)$ update time algorithm for this problem that achieves an $f$-approximation, where $f$ is the maximum number of sets that an element belongs to; under the unique games conjecture, this approximation is best possible for any fixed $f$. This is the first algorithm for dynamic set cover with approximation ratio that {exactly} matches $f$ (as opposed to {almost} $f$ in prior work), as well as the first one with runtime \emph{independent of $n,m$} (for any approximation factor of $o(f^3)$). Prior to our work, the state-of-the-art algorithms for this problem were $O(f^2)$ update time algorithms of Gupta et al. [STOC'17] and Bhattacharya et al. [IPCO'17] with $O(f^3)$ approximation, and the recent algorithm of Bhattacharya et al. [FOCS'19] with $O(f \cdot \log{n}/\epsilon^2)$ update time and $(1+\epsilon) \cdot f$ approximation, improving the $O(f^2 \cdot \log{n}/\epsilon^5)$ bound of Abboud et al. [STOC'19]. The key technical ingredient of our work is an algorithm for maintaining a {maximal} matching in a dynamic hypergraph of rank $r$, where each hyperedge has at most $r$ vertices, which undergoes hyperedge insertions and deletions in $O(r^2)$ amortized update time; our algorithm is randomized, and the bound on the update time holds in expectation and with high probability. This result generalizes the maximal matching algorithm of Solomon [FOCS'16] with constant update time in ordinary graphs to hypergraphs, and is of independent merit; the previous state-of-the-art algorithms for set cover do not translate to (integral) matchings for hypergraphs, let alone a maximal one. Our quantitative result for the set cover problem is [...]