Dynamic Set Cover with Worst-Case Recourse
Shay Solomon, Amitai Uzrad
TL;DR
This work tackles dynamic set cover under element updates, aiming to achieve fast worst-case update times with low recourse. It introduces a black-box transformation that, given any $\alpha$-approximate SC algorithm with update time $T$, yields a $(2+\varepsilon)\alpha$-approximation with update time $T+O\left(\dfrac{\alpha C}{\varepsilon}\right)$ and worst-case recourse $O\left(\dfrac{\alpha C}{\varepsilon}\right)$, enabling concrete regimes for constant aspect ratio $C$. The authors show two principal regimes: a low-frequency setting achieving $(2+\varepsilon)f$-approximation with update time $O\left(\dfrac{f\log n}{\varepsilon^2} + \dfrac{fC}{\varepsilon}\right)$ and recourse $O\left(\dfrac{fC}{\varepsilon}\right)$, and a high-frequency setting achieving $(2+\varepsilon)\ln n$-approximation with update time $O\left(\dfrac{f\log n}{\varepsilon^2} + \dfrac{C}{\varepsilon}\right)$ and recourse $O\left(\dfrac{C}{\varepsilon}\right)$. A core technical contribution is establishing a robustness property for the greedy-based dynamic SC algorithm, which is then leveraged to remove a $\log n$ factor from the recourse in the high-frequency setting. Additionally, the paper develops a static reconfiguration framework that underpins the transformation and proves approximate guarantees through careful interval-based scheduling. The results advance practical dynamic SC by balancing update-time efficiency with tight worst-case recourse, with implications for related problems such as dominating set via reductions.
Abstract
In the dynamic set cover (SC) problem, the input is a dynamic universe of at most $n$ elements and a fixed collection of $m$ sets, where each element belongs to at most $f$ sets and each set has cost in $[1/C, 1]$. The objective is to efficiently maintain an approximate minimum SC under element updates; efficiency is primarily measured by the update time, but another important parameter is the recourse (number of changes to the solution per update). Ideally, one would like to achieve low worst-case bounds on both update time and recourse. One can achieve approximation $(1+ε)\ln n$ (greedy-based) or $(1+ε)f$ (primal-dual-based) with worst-case update time $O(f\log n)$ (ignoring $ε$ dependencies). However, despite a large body of work, no algorithm with low update time (even amortized) and nontrivial worst-case recourse is known, even for unweighted instances ($C = 1$)! We remedy this by providing a transformation that, given as a black-box a SC algorithm with approximation $α$ and update time $T$, returns a set cover algorithm with approximation $(2 + ε)α$, update time $O(T + αC)$, and worst-case recourse $O(αC)$. Our main results are obtained by leveraging this transformation for constant $C$:...