Dynamic Set Cover: Improved Algorithms & Lower Bounds

TL;DR

This paper introduces new algorithms for the dynamic set cover problem that achieve near-optimal approximation ratios with efficient update times, and establishes lower bounds showing linear dependence on the frequency parameter is necessary for better approximations.

Contribution

It provides the first dynamic algorithms with approximation factors linear in the maximum set frequency, and proves lower bounds on update time and approximation trade-offs.

Findings

Achieves $(1+\epsilon)f$-approximation with $O(f^2 ext{log} n/ ext{epsilon}^5)$ update time.

Improves decremental update time to $O(f^2/ ext{epsilon}^5)$ while maintaining approximation.

Shows that faster update times imply worse approximation factors under ETH.

Abstract

We give new upper and lower bounds for the {\em dynamic} set cover problem. First, we give a $(1+\epsilon) f$-approximation for fully dynamic set cover in $O(f^2\log n /\epsilon^5)$ (amortized) update time, for any $\epsilon > 0$, where $f$ is the maximum number of sets that an element belongs to. In the decremental setting, the update time can be improved to $O(f^2/\epsilon^5)$, while still obtaining an $(1+\epsilon) f$-approximation. These are the first algorithms that obtain an approximation factor linear in $f$ for dynamic set cover, thereby almost matching the best bounds known in the offline setting and improving upon the previous best approximation of $O(f^2)$ in the dynamic setting. To complement our upper bounds, we also show that a linear dependence of the update time on $f$ is necessary unless we can tolerate much worse approximation factors. Using the recent distributed PCP-framework, we show that any dynamic set cover algorithm that has an amortized update time of $O(f^{1-\epsilon})$ must have an approximation factor that is $\Omega(n^\delta)$ for some constant $\delta>0$ under the Strong Exponential Time Hypothesis.