Scaling and Proximity Properties of Integrally Convex Functions

Authors Satoko Moriguchi, Kazuo Murota, Akihisa Tamura, Fabio Tardella

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Satoko Moriguchi
Kazuo Murota
Akihisa Tamura
Fabio Tardella

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Satoko Moriguchi, Kazuo Murota, Akihisa Tamura, and Fabio Tardella. Scaling and Proximity Properties of Integrally Convex Functions. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 57:1-57:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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.
  • Discrete optimization
  • discrete convexity
  • proximity theorem
  • scaling algorithm


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