A note on the budgeted maximization of submodular functions

Many set functions F in combinatorial optimization satisfy the diminishing returns property F{A U X) — F(A) > F(A' U X) — F(A') for A C A!. Such functions are called submodular. A result from Nemhauser etal. states that the problem of selecting ^-element subsets maximizing a nondecreasing submodular function can be approximated with a constant factor (1 — 1/e) performance guarantee. Khuller etal. showed that for the special submodular function involved in the MAX-COVER problem, this approximation result generalizes to a budgeted setting under linear nonnegative cost-functions. In this note, we extend this result to general submodular functions. Motivated by the problem of maximizing entropy in discrete graphical models, where the submodular objective cannot be evaluated exactly, we generalize our result to account for absolute errors. University L i ' . ^ne* Carnegie Mellon 11 * : v Pittsburgh, PA 15;