Characterizing discrete functions that extend through local convex relaxations
A Characterization of Functions on the Integer Lattice That Are L-Extendable through Local Convex Extensions
Discrete Mathematics
Summary
The paper studies special functions defined on whole-number grids that can be smoothly extended to nearby half-integers while keeping certain convexity properties. The authors find that such functions must be integrally convex and identify a specific kind where their 'local convex extension' works well for optimization. They describe this subclass precisely using a concept called D-convexity, which involves how function values relate within small unit cubes. This work helps clarify the structure and optimization of these discrete functions.
What this means in practice
- •For optimization engineers: Design discrete optimization methods that guarantee matching optimal values using local convex extensions for problems on integer grids.
- •For data scientists in combinatorial optimization: Employ the characterization of D-convex sets to better model and solve integer optimization problems involving unit hypercube constraints.
A theory result. No direct application yet.
Authors
Ken Yokoyama, Yuni Iwamasa, Kei Kimura
Abstract
L-extendable functions on the integer lattice are functions that admit a $\frac12$UJ-convex relaxation; that is, values can be assigned to the additional half-integral points, while keeping the original values at integer points, so that the resulting function is $\frac12$UJ-convex. Various examples of L-extendable functions and their applications are known. A general characterization of L-extendability for functions on the integer lattice, however, remains open. In this paper, we study the L-extendability of functions on the integer lattice. We first obtain a necessary condition: L-extendable functions are integrally convex. We then consider a special subclass of L-extendable functions, namely, those whose local convex extensions give L-convex relaxations. This subclass has a useful optimization property: the optimal value of the original function coincides with that of the $\frac12$UJ-convex relaxation given by its local convex extension, a property not guaranteed for general $\frac12$UJ-convex relaxations. We characterize this subclass by showing that a function is L-extendable through its local convex extension if and only if it is integrally convex and its restriction to every unit hypercube is D-convex, where D-convexity is a discrete convexity associated with reflected Freudenthal simplicial divisions. This subclass strictly contains the class of D-convex functions. We also give a combinatorial characterization of D-convex subsets of the vertices of a unit hypercube: a set $X\subseteq\{0,1\}^n$ is D-convex if and only if it is majority-closed and contains a relative opposite pair.