← 研踪 PaperTrailarXiv:2610.38287 · v1
A Characterization of Functions on the Integer Lattice That Are L-Extendable through Local Convex Extensions
首次提交 2026-10-28 13:12
版本更新 2026-10-28 13:12
雷达观测 2026-10-30 10:22
摘要
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.
作者备注：17 pages, 3 figures
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v1 · 2026-10-28 13:12 · 最新观测版本
A Characterization of Functions on the Integer Lattice That Are L-Extendable through Local Convex Extensions
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.
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