Approximating Star Cover Problems

TL;DR

This paper introduces new bicriteria approximation algorithms for star cover problems in metric spaces, improving upon previous results by using novel LP rounding techniques.

Contribution

The authors develop novel LP rounding algorithms that provide bicriteria approximations for minimum load and minimum size star cover problems, surpassing prior work.

Findings

Achieved a $(1+ ext{epsilon})k$ star cover with bounded load.

Provided a star cover with $O(1/ ext{epsilon}^2)$ times the optimal number of stars.

Extended approximation results to general metric spaces, not just when $F=C$.

Abstract

Given a metric space $(F \cup C, d)$, we consider star covers of $C$ with balanced loads. A star is a pair $(f, C_f)$ where $f \in F$ and $C_f \subseteq C$, and the load of a star is $\sum_{c \in C_f} d(f, c)$. In minimum load $k$-star cover problem $(\mathrm{MLkSC})$, one tries to cover the set of clients $C$ using $k$ stars that minimize the maximum load of a star, and in minimum size star cover $(\mathrm{MSSC})$ one aims to find the minimum number of stars of load at most $T$ needed to cover $C$, where $T$ is a given parameter. We obtain new bicriteria approximations for the two problems using novel rounding algorithms for their standard LP relaxations. For $\mathrm{MLkSC}$, we find a star cover with $(1+\varepsilon)k$ stars and $O(1/\varepsilon^2)\mathrm{OPT}_{\mathrm{MLk}}$ load where $\mathrm{OPT}_{\mathrm{MLk}}$ is the optimum load. For $\mathrm{MSSC}$, we find a star cover with $O(1/\varepsilon^2) \mathrm{OPT}_{\mathrm{MS}}$ stars of load at most $(2 + \varepsilon) T$ where $\mathrm{OPT}_{\mathrm{MS}}$ is the optimal number of stars for the problem. Previously, non-trivial bicriteria approximations were known only when $F = C$.