A Proportionally Submodular Functions

Submodular functions are well-studied in combinatorial optimization, game theory and economics. The natural diminishing returns property makes them suitable for many applications. We study an extension of monotone submodular functions, which we call proportionally submodular functions. Our extension includes some (mildly) supermodular functions. We show that several natural functions belong to this class and relate our class to some other recent submodular function extensions. We consider the optimization problem of maximizing a proportionally submodular function subject to uniform and general matroid constraints. For a uniform matroid constraint, the “standard greedy algorithm” achieves a constant approximation ratio. More specifically, for any cardinality constraint p, the greedy algorithm has a constant approximation ratio bounded by a function α(p) that experimentally appears to be converging (from below) to 5.95 as p increases. For a general matroid constraint with rank s, we prove that the local search algorithm has constant approximation ratio bounded by a function ρ(s) which analytically is converging (from above) to 10.22 as s increases.