Dynamic Set Cover with Worst-Case Recourse

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 a 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 (the number of changes to solution per update). Ideally, one would like to achieve low worst-case bounds on both update time and recourse.
One can achieve an approximation of (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 a SC algorithm with approximation α and update time T as a black-box, returns a set cover algorithm with approximation (2 + ε)α, update time O(T + α C) and worst-case recourse O(α C). Our main results are obtained by leveraging this transformation for constant C:  
- For f = O(log n), applying the transformation on the best primal-dual-based algorithm yields worst-case recourse O(f). For constant f (e.g., vertex cover), we get near-optimal bounds on all parameters. 
- For f = Ω(log n), applying the transformation on the best greedy-based algorithm yields worst-case recourse O(log n). As our main technical contribution, we show that by opening the black box and exploiting a certain robustness property of the greedy-based algorithm, the worst-case recourse can be reduced to O(1), without sacrificing the other parameters, yielding a ((2 + ε) ln n)-approximation with worst-case update time O(flog n) and O(1) worst-case recourse.