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The Set Cover problem (SCP) and Set Packing problem (SPP) are standardNP-hard combinatorial optimization problems. Their decision problem versionsare shown to be NP-Complete in Karp's 1972 paper. We specify a rough guide toconstructing approximation heuristics that may have widespread applications andapply it to devise greedy approximation algorithms for SCP and SPP, where theselection heuristic is a variation of that in the standard greedy approximationalgorithm. Our technique involves assigning to each input set a valuation andthen selecting, in each round, the set whose valuation is highest. We provethat the technique we use for determining a valuation of the input sets yieldsa unique value for all Set Cover instances. For both SCP and SPP we giveexperimental evidence that the valuations we specify are unique and can becomputed to high precision quickly by an iterative algorithm. Others haveexperimented with testing the observed approximation ratio of variousalgorithms over a variety of randomly generated instances, and we haveextensive experimental evidence to show the quality of the new algorithmrelative to greedy heuristics in common use. Our algorithms are somewhat morecomputationally intensive than the standard heuristics, though they are stillpractical for large instances. We discuss some ways to speed up our algorithmsthat do not significantly distort their effectiveness in practice on randominstances.

New Greedy Heuristics For Set Cover and Set Packing