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

Kovaleva, Sofia; Spieksma, Frits C. R.

RAIRO - Operations Research - Recherche Opérationnelle, Tome 36 (2002) no. 1, pp. 53-71.

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 $𝒫=\mathrm{𝒩𝒫}$, the covering problem cannot be approximated in polynomial time within arbitrarily good precision.

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