A Characterization of Functions on the Integer Lattice That Are L-Extendable through Local Convex Extensions
Abstract
L-extendable functions on the integer lattice are functions that admit a 1/2UJ-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 1/2UJ-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 1/2UJ-convex relaxation given by its local convex extension, a property not guaranteed for general 1/2UJ-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⊆{0,1}n is D-convex if and only if it is majority-closed and contains a relative opposite pair.
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BibTeX
@misc{yokoyama2026characterization,
title = {{A Characterization of Functions on the Integer Lattice That Are L-Extendable through Local Convex Extensions}},
author = {Ken Yokoyama and Yuni Iwamasa and Kei Kimura},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38287},
eprint = {2610.38287},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38287},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Characterization of Functions on the Integer Lattice That Are L-Extendable through Local Convex Extensions
AU - Ken Yokoyama
AU - Yuni Iwamasa
AU - Kei Kimura
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38287
UR - https://arxiv.org/abs/2610.38287
AB - L-extendable functions on the integer lattice are functions that admit a 1/2UJ-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 1/2UJ-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 1/2UJ-convex relaxation given by its local convex extension, a property not guaranteed for general 1/2UJ-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⊆{0,1}^n is D-convex if and only if it is majority-closed and contains a relative opposite pair.
ER -