Exact and Approximation Algorithms for Geometric and Capacitated Set Cover Problems with Applications

TL;DR

This paper presents new polynomial-time exact algorithms for geometric set cover problems and develops approximation algorithms for capacitated set cover, including a PTAS for specific dual problems involving line intervals and arcs.

Contribution

It introduces the first polynomial-time exact solutions for geometric set cover variants and extends approximation techniques to capacitated set cover problems, achieving a ratio of 2.357.

Findings

First polynomial-time exact solutions for geometric set cover variants.

Approximation algorithm for capacitated set cover with ratio 2.357.

A PTAS for the dual problem with fixed number of sets.

Abstract

First, we study geometric variants of the standard set cover motivated by assignment of directional antenna and shipping with deadlines, providing the first known polynomial-time exact solutions. Next, we consider the following general capacitated set cover problem. There is given a set of elements with real weights and a family S of sets of elements. One can use a set if it is a subset of one of the sets on our lists and the sum of weights is at most one. The goal is to cover all the elements with the allowed sets.<br>We show that any polynomial-time algorithm that approximates the un-capacitated version of the set cover problem with ratio r can be converted to an approximation algorithm for the capacitated version with ratio r + 1.357.In particular, the composition of these two results yields a polynomial-time approximation algorithm for the problem of covering a set of customers represented by a weighted n-point set with a minimum number of antennas of variable angular range and fixed capacity with ratio 2.357. Finally, we provide a PTAS for the dual problem where the number of sets (e.g., antennas) to use is fixed and the task is to minimize the maximum set load, in case the sets correspond to line intervals or arcs.