New query lower bounds for submodular function minimization

We consider submodular function minimization in the oracle model: given black-box access to a submodular set function f : 2^[n^] → R, find an element of arg minS{f(S)} using as few queries to f(·) as possible. State-of-the-art algorithms succeed with Õ(n²) queries [13], yet the best-known lower bound has never been improved beyond n [6]. We provide a query lower bound of 2n for submodular function minimization, a 3n/2 − 2 query lower bound for the non-trivial minimizer of a symmetric submodular function, and a ⁿ₂ query lower bound for the non-trivial minimizer of an asymmetric submodular function. Our 3n/2 − 2 lower bound results from a connection between SFM lower bounds and a novel concept we term the cut dimension of a graph. Interestingly, this yields a 3n/2 − 2 cut-query lower bound for finding the global mincut in an undirected, weighted graph, but we also prove it cannot yield a lower bound better than n + 1 for s-t mincut, even in a directed, weighted graph.