On the Partition Set Cover Problem

TL;DR

This paper introduces a new approximation algorithm for the Partition Set Cover problem, extending known techniques to handle partitioned elements with guarantees related to the number of partitions and existing set cover approximations.

Contribution

It presents a randomized LP-rounding algorithm achieving an $O(eta + ext{log } r)$ approximation for the Partition Set Cover problem, improving guarantees for related set systems.

Findings

The algorithm achieves an $O(eta + ext{log } r)$ approximation.

Improved guarantees for set systems with sublogarithmic $eta$.

NP-hardness of approximating within $o( ext{log } r)$.

Abstract

Several algorithms with an approximation guarantee of $O(\log n)$ are known for the Set Cover problem, where $n$ is the number of elements. We study a generalization of the Set Cover problem, called the Partition Set Cover problem. Here, the elements are partitioned into $r$ \emph{color classes}, and we are required to cover at least $k_t$ elements from each color class $\mathcal{C}_t$, using the minimum number of sets. We give a randomized LP-rounding algorithm that is an $O(\beta + \log r)$ approximation for the Partition Set Cover problem. Here $\beta$ denotes the approximation guarantee for a related Set Cover instance obtained by rounding the standard LP. As a corollary, we obtain improved approximation guarantees for various set systems for which $\beta$ is known to be sublogarithmic in $n$. We also extend the LP rounding algorithm to obtain $O(\log r)$ approximations for similar generalizations of the Facility Location type problems. Finally, we show that many of these results are essentially tight, by showing that it is NP-hard to obtain an $o(\log r)$-approximation for any of these problems.