License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2020.64
URN: urn:nbn:de:0030-drops-117493
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/11749/
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Graur, Andrei ; Pollner, Tristan ; Ramaswamy, Vidhya ; Weinberg, S. Matthew

New Query Lower Bounds for Submodular Function Minimization

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Abstract

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 min_S {f(S)} using as few queries to f(⋅) as possible. State-of-the-art algorithms succeed with Õ(n²) queries [Yin Tat Lee et al., 2015], yet the best-known lower bound has never been improved beyond n [Nicholas J. A. Harvey, 2008].
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.

BibTeX - Entry

@InProceedings{graur_et_al:LIPIcs:2020:11749,
  author =	{Andrei Graur and Tristan Pollner and Vidhya Ramaswamy and S. Matthew Weinberg},
  title =	{{New Query Lower Bounds for Submodular Function Minimization}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{64:1--64:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Thomas Vidick},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/11749},
  URN =		{urn:nbn:de:0030-drops-117493},
  doi =		{10.4230/LIPIcs.ITCS.2020.64},
  annote =	{Keywords: submodular functions, query lower bounds, min cut}
}

Keywords: submodular functions, query lower bounds, min cut
Collection: 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)
Issue Date: 2020
Date of publication: 06.01.2020

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