Randomized Online Algorithms for Set Cover Leasing Problems

Abstract

In the leasing variant of Set Cover presented by Anthony et al. [1], elements $U$ arrive over time and must be covered by sets from a family $F$ of subsets of $U$. Each set can be leased for $K$ different periods of time. Let $\left| U \right| = n$ and $\left| F \right| = m$. Leasing a set $S$ for a period $k$ incurs a cost $c_{S}^{k}$ and allows $S$ to cover its elements for the next $l_k$ time steps. The objective is to minimize the total cost of the sets leased, such that elements arriving at any time $t$ are covered by sets which contain them and are leased during time $t$. Anthony et al. [1] gave an optimal $O(\log n)$-approximation for the problem in the offline setting, unless $\mathcal {P} = \mathcal {NP}$ [22]. In this paper, we give randomized algorithms for variants of Set Cover Leasing in the online setting, including a generalization of Online Set Cover with Repetitions presented by Alon et al. [2], where elements appear multiple times and must be covered by a different set at each arrival. Our results improve the $\mathcal {O}(\log ^2 (mn))$ competitive factor of Online Set Cover with Repetitions [2] to $\mathcal {O}(\log d \log (dn)) = \mathcal {O}(\log m \log (mn))$, where $d$ is the maximum number of sets an element belongs to.

This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Centre “On-The-Fly Computing” (SFB 901) and the International Graduate School “Dynamic Intelligent Systems”.