  We introduce a parameterized version of set cover that generalizes several
previously studied problems. Given a ground set V and a collection of subsets
S_i of V, a feasible solution is a partition of V such that each subset of the
partition is included in one of the S_i. The problem involves maximizing the
mean subset size of the partition, where the mean is the generalized mean of
parameter p, taken over the elements. For p=-1, the problem is equivalent to
the classical minimum set cover problem. For p=0, it is equivalent to the
minimum entropy set cover problem, introduced by Halperin and Karp. For p=1,
the problem includes the maximum-edge clique partition problem as a special
case. We prove that the greedy algorithm simultaneously approximates the
problem within a factor of (p+1)^1/p for any p in R^+, and that this is the
best possible unless P=NP. These results both generalize and simplify previous
results for special cases. We also consider the corresponding graph coloring
problem, and prove several tractability and inapproximability results. Finally,
we consider a further generalization of the set cover problem in which we aim
at minimizing the sum of some concave function of the part sizes. As an
application, we derive an approximation ratio for a Rent-or-Buy set cover
problem.
