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 \(f:2^{[n]}\rightarrow \mathbb{R}\), find an element of \(\arg\min_S \{f(S)\}\) using as few queries to \(f(\cdot)\) as possible. State-of-the-art algorithms succeed with \(\tilde{O}...

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Bibliographic Details
Published inarXiv.org
Main Authors Graur, Andrei, Pollner, Tristan, Ramaswamy, Vidhya, Weinberg, S Matthew
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 15.11.2019
Subjects
Algorithms
Lower bounds
Optimization
Production scheduling
Queries
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Summary:We consider submodular function minimization in the oracle model: given black-box 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. State-of-the-art algorithms succeed with \(\tilde{O}(n^2)\) queries [LeeSW15], yet the best-known lower bound has never been improved beyond \(n\) [Harvey08]. 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 \(\binom{n}{2}\) 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.
ISSN:2331-8422