Combinatorial Interpretations of Dual Fitting and Primal Fitting

We present two new combinatorial approximation frameworks that are not based on LPduality, or even on linear programming. Instead, they are based on weight manipulation in the spirit of the local ratio technique. We show that the first framework is equivalent to the LP based method of dual fitting and that the second framework is equivalent to an LP-based method which we define and call primal fitting. Our equivalence results are similar to the proven equivalence between the (combinatorial) local ratio technique and the (LP based) primal-dual schema. We demonstrate the frameworks by presenting alternative analyses to the greedy algorithm for the set cover problem, and to known algorithms for the metric uncapacitated facility location problem. We use our second framework to design an approximation algorithm for the multiple metric uncapacitated facility location problem. We also analyze a 9-approximation algorithm for a disk cover problem that arises in the design phase of cellular telephone networks. Finally, we present constant factor approximation algorithms for the prize collecting and partial versions of this disk cover problem.