Primal-dual approximation algorithms for a packing-covering pair of problems

¹ Department of Mathematics, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands; s.kovaleva@math.unimaas.nl.
² Department of Applied Economics, Katholieke Universiteit Leuven, Naamsestraat 69, 3000 Leuven, Belgium.

Received: December 2000

Abstract

We consider a special packing-covering pair of problems. The packing problem is a natural generalization of finding a (weighted) maximum independent set in an interval graph, the covering problem generalizes the problem of finding a (weighted) minimum clique cover in an interval graph. The problem pair involves weights and capacities; we consider the case of unit weights and the case of unit capacities. In each case we describe a simple algorithm that outputs a solution to the packing problem and to the covering problem that are within a factor of 2 of each other. Each of these results implies an approximative min-max result. For the general case of arbitrary weights and capacities we describe an LP-based (2 + ε)-approximation algorithm for the covering problem. Finally, we show that, unless P = NP, the covering problem cannot be approximated in polynomial time within arbitrarily good precision.