Maximizing k -Submodular Functions and Beyond

Abstract

We consider the maximization problem in the value oracle model of functions defined on k -tuples of sets that are submodular in every orthant and r -wise monotone, where k ⩾ 2 and 1 ⩽ r ⩽ k . We give an analysis of a deterministic greedy algorithm that shows that any such function can be approximated to a factor of 1/(1 + r ). For r = k , we give an analysis of a randomized greedy algorithm that shows that any such function can be approximated to a factor of 1/(1+√ k /2. In the case of k = r = 2, the considered functions correspond precisely to bisubmodular functions, in which case we obtain an approximation guarantee of 1/2. We show that, as in the case of submodular functions, this result is the best possible both in the value query model and under the assumption that NP ≠ RP . Extending a result of Ando et al., we show that for any k ⩾ 3, submodularity in every orthant and pairwise monotonicity (i.e., r = 2) precisely characterize k -submodular functions. Consequently, we obtain an approximation guarantee of 1/3 (and thus independent of k ) for the maximization problem of k -submodular functions.