Approximate Covering with Lower and Upper Bounds via LP Rounding

TL;DR

This paper introduces an LP rounding approach to approximate the lower- and upper-bounded covering problem, achieving constant-factor solutions with small violations of bounds, extending previous work limited to upper bounds.

Contribution

It presents the first LP rounding algorithm for the combined lower- and upper-bounded covering problem with provable approximation guarantees.

Findings

Achieves constant approximation with small bound violations.

Extends known results from upper-bound-only covering to combined bounds.

Provides bounds violation-free solutions for lower-bound-only covering.

Abstract

In this paper, we study the lower- and upper-bounded covering (LUC) problem, where we are given a set $P$ of $n$ points, a collection $\mathcal{B}$ of balls, and parameters $L$ and $U$. The goal is to find a minimum-sized subset $\mathcal{B}'\subseteq \mathcal{B}$ and an assignment of the points in $P$ to $\mathcal{B}'$, such that each point $p\in P$ is assigned to a ball that contains $p$ and for each ball $B_i\in \mathcal{B}'$, at least $L$ and at most $U$ points are assigned to $B_i$. We obtain an LP rounding based constant approximation for LUC by violating the lower and upper bound constraints by small constant factors and expanding the balls by again a small constant factor. Similar results were known before for covering problems with only the upper bound constraint. We also show that with only the lower bound constraint, the above result can be obtained without any lower bound violation. Covering problems have close connections with facility location problems. We note that the known constant-approximation for the corresponding lower- and upper-bounded facility location problem, violates the lower and upper bound constraints by a constant factor.