Fully Dynamic Set Cover: Worst-Case Recourse and Update Time
Sayan Bhattacharya, Ruoxu Cen, Debmalya Panigrahi
TL;DR
This work resolves long-standing open questions for dynamic set cover by giving the first fully dynamic algorithms with non-trivial worst-case recourse and update-time guarantees in both approximation regimes. The authors introduce a hierarchical, level-based framework and a three-solution architecture (foreground, background, and buffers) to tightly bound recourse while preserving feasibility and approximation quality. A catch-up greedy technique and a de-amortization reduction reduce insertion recourse, enabling near-optimal worst-case guarantees: an O(log n)-approximation with O(log n) recourse and Õ(f) update time, and an O(f)-approximation with Õ(1) recourse and Õ(f) update time. The results apply to the unweighted setting and lay groundwork for potential extensions to weighted instances, with practical implications for systems requiring stable, efficient dynamic set covers.
Abstract
In (fully) dynamic set cover, the goal is to maintain an approximately optimal solution to a dynamically evolving instance of set cover, where in each step either an element is added to or removed from the instance. The two main desiderata of a dynamic set cover algorithm are to minimize at each time-step, the recourse, which is the number of sets removed from or added to the solution, and the update time to compute the updated solution. This problem has been extensively studied over the last decade leading to many results that achieve ever-improving bounds on the recourse and update time, while maintaining a solution whose cost is comparable to that of offline approximation algorithms. In this paper, we give the first algorithms to simultaneously achieve non-trivial worst-case bounds for recourse and update time. Specifically, we give fully-dynamic set cover algorithms that simultaneously achieve $O(\log n)$ recourse and $f\cdot \textrm{poly}\log(n)$ update time in the worst-case, for both approximation regimes: $O(\log n)$ and $O(f)$ approximation. (Here, $n, f$ respectively denote the maximum number of elements and maximum frequency of an element across all instances.) Prior to our work, all results for this problem either settled for amortized bounds on recourse and update time, or obtained $f\cdot \textrm{poly}\log(n)$ update time in the worst-case but at the cost of $Ω(m)$ worst-case recourse. (Here, $m$ denotes the number of sets. Note that any algorithm has recourse at most $m$.)