# Randomized Online Algorithms for Set Cover Leasing Problems

• Sebastian Abshoff
• Christine Markarian
• Friedhelm Meyer auf der Heide
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8881)

## 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.

## Keywords

Set cover Multicover Online algorithms Randomized algorithms Leasing

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© Springer International Publishing Switzerland 2014

## Authors and Affiliations

• Sebastian Abshoff
• 1
• Christine Markarian
• 1
Email author
• Friedhelm Meyer auf der Heide
• 1
1. 1.Computer Science Department, Heinz Nixdorf InstituteUniversity of PaderbornPaderbornGermany