  The Set Cover problem (SCP) and Set Packing problem (SPP) are standard
NP-hard combinatorial optimization problems. Their decision problem versions
are shown to be NP-Complete in Karp's 1972 paper. We specify a rough guide to
constructing approximation heuristics that may have widespread applications and
apply it to devise greedy approximation algorithms for SCP and SPP, where the
selection heuristic is a variation of that in the standard greedy approximation
algorithm. Our technique involves assigning to each input set a valuation and
then selecting, in each round, the set whose valuation is highest. We prove
that the technique we use for determining a valuation of the input sets yields
a unique value for all Set Cover instances. For both SCP and SPP we give
experimental evidence that the valuations we specify are unique and can be
computed to high precision quickly by an iterative algorithm. Others have
experimented with testing the observed approximation ratio of various
algorithms over a variety of randomly generated instances, and we have
extensive experimental evidence to show the quality of the new algorithm
relative to greedy heuristics in common use. Our algorithms are somewhat more
computationally intensive than the standard heuristics, though they are still
practical for large instances. We discuss some ways to speed up our algorithms
that do not significantly distort their effectiveness in practice on random
instances.
