Hard Capacitated Set Cover and Uncapacitated Geometric Set Cover

TL;DR

This paper improves approximation algorithms for capacitated set cover and geometric hitting set problems, providing simpler analyses and tighter bounds, including polynomial-time epsilon-net construction and logarithmic approximation factors.

Contribution

It offers a simpler, more intuitive analysis for logarithmic approximation of capacitated set cover and extends epsilon-net construction to geometric hitting sets with improved bounds.

Findings

Logarithmic approximation factor achieved for capacitated set cover.

Polynomial-time construction of small epsilon-nets for geometric hitting set.

Hitting set size bounded by O(log log OPT) for axis-parallel rectangles.

Abstract

The first part of this report describes the following result that, logarithmic approximation factor for hard capacitated set cover can be achieved from Wolsey's work [9], using a simpler and more intuitive analysis. We further show in our work, that O(log n) approximation factor can be achieved for the same problem by applying analysis of general set cover to analyze Wolsey's algorithm [5]. This work is based on the key observation that we make in Lemma 3 of this report. The second part of the report describes the geometric hitting set problem, where X is a ground set of points in a plane and S is a set of axis parallel rectangles. It is shown that epsilon-nets of size O(1/epsilon log log 1/epsilon) can be computed in polynomial time. Applying Bronnimann and Goodrich result [3] gives the hitting set of size O(log log OPT) for this problem.