Scaling and proximity properties of integrally convex functions

Satoko Moriguchi, Kazuo Murota, Akihisa Tamura, Fabio Tardella

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

Abstract

In discrete convex analysis, the scaling and proximity properties for the class of L-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 ≤ 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-convex and integrally convex functions but enjoys the same scaling and proximity properties that hold for L-convex functions.

Original languageEnglish
Title of host publication27th International Symposium on Algorithms and Computation, ISAAC 2016
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages57.1-57.13
Volume64
ISBN (Electronic)9783959770262
DOIs
Publication statusPublished - 2016 Dec 1
Event27th International Symposium on Algorithms and Computation, ISAAC 2016 - Sydney, Australia
Duration: 2016 Dec 122016 Dec 14

Other

Other27th International Symposium on Algorithms and Computation, ISAAC 2016
CountryAustralia
CitySydney
Period16/12/1216/12/14

Keywords

  • Discrete convexity
  • Discrete optimization
  • Proximity theorem
  • Scaling algorithm

ASJC Scopus subject areas

  • Software

Cite this

Moriguchi, S., Murota, K., Tamura, A., & Tardella, F. (2016). Scaling and proximity properties of integrally convex functions. In 27th International Symposium on Algorithms and Computation, ISAAC 2016 (Vol. 64, pp. 57.1-57.13). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ISAAC.2016.57

Scaling and proximity properties of integrally convex functions. / Moriguchi, Satoko; Murota, Kazuo; Tamura, Akihisa; Tardella, Fabio.

27th International Symposium on Algorithms and Computation, ISAAC 2016. Vol. 64 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2016. p. 57.1-57.13.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Moriguchi, S, Murota, K, Tamura, A & Tardella, F 2016, Scaling and proximity properties of integrally convex functions. in 27th International Symposium on Algorithms and Computation, ISAAC 2016. vol. 64, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, pp. 57.1-57.13, 27th International Symposium on Algorithms and Computation, ISAAC 2016, Sydney, Australia, 16/12/12. https://doi.org/10.4230/LIPIcs.ISAAC.2016.57
Moriguchi S, Murota K, Tamura A, Tardella F. Scaling and proximity properties of integrally convex functions. In 27th International Symposium on Algorithms and Computation, ISAAC 2016. Vol. 64. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2016. p. 57.1-57.13 https://doi.org/10.4230/LIPIcs.ISAAC.2016.57
Moriguchi, Satoko ; Murota, Kazuo ; Tamura, Akihisa ; Tardella, Fabio. / Scaling and proximity properties of integrally convex functions. 27th International Symposium on Algorithms and Computation, ISAAC 2016. Vol. 64 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2016. pp. 57.1-57.13
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