On Column-restricted and Priority Covering Integer Programs

TL;DR

This paper studies column-restricted covering integer programs (CCIPs) and their relation to 0-1-CIPs, providing approximation algorithms based on integrality gaps, and investigates priority versions of line and tree cover problems.

Contribution

It establishes a connection between CCIP approximability and the integrality gaps of related 0-1-CIPs and priority variants, introducing new algorithms and hardness results.

Findings

Provides an O(gamma + omega) approximation algorithm for CCIPs.

Gives a polynomial-time exact algorithm for the priority line cover problem.

Shows the priority tree cover problem is APX-hard and offers a 2-approximation algorithm.

Abstract

In a column-restricted covering integer program (CCIP), all the non-zero entries of any column of the constraint matrix are equal. Such programs capture capacitated versions of covering problems. In this paper, we study the approximability of CCIPs, in particular, their relation to the integrality gaps of the underlying 0,1-CIP. If the underlying 0,1-CIP has an integrality gap O(gamma), and assuming that the integrality gap of the priority version of the 0,1-CIP is O(omega), we give a factor O(gamma + omega) approximation algorithm for the CCIP. Priority versions of 0,1-CIPs (PCIPs) naturally capture quality of service type constraints in a covering problem. We investigate priority versions of the line (PLC) and the (rooted) tree cover (PTC) problems. Apart from being natural objects to study, these problems fall in a class of fundamental geometric covering problems. We bound the integrality of certain classes of this PCIP by a constant. Algorithmically, we give a polytime exact algorithm for PLC, show that the PTC problem is APX-hard, and give a factor 2-approximation algorithm for it.