More on Disjunctive Convex Hulls of Generalized Cross Polytopes
Abstract
We continue the study of the convex hull D⊂ℝd+n of a disjunction of n+1 polytopes in ℝd, managed by n binary indicator variables. Previously, D had been characterized for arbitrary polytopes when d∈{1,2}, for polytopes sharing a common constraint matrix (under a technical condition, including hyper-rectangles), and for two generalized cross polytopes (n=1). Here we give a complete description of D for arbitrary n and d when all n+1 polytopes are generalized cross polytopes. Every vertical facet is described by a normalized support inequality, and such an inequality is facet describing if and only if its equality graph, a bipartite graph on the polytopes and the coordinates, is connected; all vertical facets are generated explicitly from feasible spanning trees. We characterize when such a facet is one of the previously known pairwise inequalities, and we show that pairwise inequalities suffice when d≤3, but not in general. We establish a bijection between the vertical facets and the maximal cells of a regular subdivision of Δn×Δd−1\,, which yields the sharp bound n+1+2dbinomn+d−1n on the number of facets of D, attained for generic data. We also give a polynomial-time separation algorithm that returns a violated facet, which is maximally violated among the normalized support inequalities. Computational experiments show that the facets that are not pairwise inequalities close a large part of the gap left by the pairwise inequalities.
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BibTeX
@misc{qu2026more,
title = {{More on Disjunctive Convex Hulls of Generalized Cross Polytopes}},
author = {Yushan Qu and Jon Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05560},
eprint = {2610.05560},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05560},
}
RIS
TY - UNPB
TI - More on Disjunctive Convex Hulls of Generalized Cross Polytopes
AU - Yushan Qu
AU - Jon Lee
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05560
UR - https://arxiv.org/abs/2610.05560
AB - We continue the study of the convex hull D⊂ℝ^d+n of a disjunction of n+1 polytopes in ℝ^d, managed by n binary indicator variables. Previously, D had been characterized for arbitrary polytopes when d∈{1,2}, for polytopes sharing a common constraint matrix (under a technical condition, including hyper-rectangles), and for two generalized cross polytopes (n=1). Here we give a complete description of D for arbitrary n and d when all n+1 polytopes are generalized cross polytopes. Every vertical facet is described by a normalized support inequality, and such an inequality is facet describing if and only if its equality graph, a bipartite graph on the polytopes and the coordinates, is connected; all vertical facets are generated explicitly from feasible spanning trees. We characterize when such a facet is one of the previously known pairwise inequalities, and we show that pairwise inequalities suffice when d≤3, but not in general. We establish a bijection between the vertical facets and the maximal cells of a regular subdivision of Δ_n×Δ_d−1\,, which yields the sharp bound n+1+2^dbinomn+d−1n on the number of facets of D, attained for generic data. We also give a polynomial-time separation algorithm that returns a violated facet, which is maximally violated among the normalized support inequalities. Computational experiments show that the facets that are not pairwise inequalities close a large part of the gap left by the pairwise inequalities.
ER -