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

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.