Scaling and Proximity Properties of Integrally Convex Functions
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
57:1-57:13
Regular Paper
Satoko
Moriguchi
Satoko Moriguchi
Kazuo
Murota
Kazuo Murota
Akihisa
Tamura
Akihisa Tamura
Fabio
Tardella
Fabio Tardella
10.4230/LIPIcs.ISAAC.2016.57
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode