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.ISAAC.2016.57
URN: urn:nbn:de:0030-drops-68368
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6836/
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Moriguchi, Satoko ; Murota, Kazuo ; Tamura, Akihisa ; Tardella, Fabio

Scaling and Proximity Properties of Integrally Convex Functions

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Abstract

In discrete convex analysis, the scaling and proximity properties for the class of L^natural-convex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of n variables, we show here that the scaling property only holds when n leq 2, while a proximity theorem can be established for any n, but only with an exponential bound. This is, however, sufficient to extend the classical logarithmic complexity result for minimizing a discretely convex function in one dimension to the case of integrally convex functions in two dimensions. Furthermore, we identified a new class of discrete convex functions, called directed integrally convex functions, which is strictly between the classes of L^natural -convex and integrally convex functions but enjoys the same scaling and proximity properties that hold for L^natural -convex functions.

BibTeX - Entry

@InProceedings{moriguchi_et_al:LIPIcs:2016:6836,
  author =	{Satoko Moriguchi and Kazuo Murota and Akihisa Tamura and Fabio Tardella},
  title =	{{Scaling and Proximity Properties of Integrally Convex Functions}},
  booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
  pages =	{57:1--57:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-026-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{64},
  editor =	{Seok-Hee Hong},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6836},
  URN =		{urn:nbn:de:0030-drops-68368},
  doi =		{10.4230/LIPIcs.ISAAC.2016.57},
  annote =	{Keywords: Discrete optimization, discrete convexity, proximity theorem, scaling algorithm}
}

Keywords: Discrete optimization, discrete convexity, proximity theorem, scaling algorithm
Collection: 27th International Symposium on Algorithms and Computation (ISAAC 2016)
Issue Date: 2016
Date of publication: 07.12.2016

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