New Query Lower Bounds for Submodular Function MInimization
Abstract
We consider submodular function minimization in the oracle model: given blackbox access to a submodular set function $f:2^{[n]}\rightarrow \mathbb{R}$, find an element of $\arg\min_S \{f(S)\}$ using as few queries to $f(\cdot)$ as possible. Stateoftheart algorithms succeed with $\tilde{O}(n^2)$ queries [LeeSW15], yet the bestknown lower bound has never been improved beyond $n$ [Harvey08]. We provide a query lower bound of $2n$ for submodular function minimization, a $3n/22$ query lower bound for the nontrivial minimizer of a symmetric submodular function, and a $\binom{n}{2}$ query lower bound for the nontrivial minimizer of an asymmetric submodular function. Our $3n/22$ 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 $3n/22$ cutquery 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.06889
 Bibcode:
 2019arXiv191106889G
 Keywords:

 Computer Science  Data Structures and Algorithms