Eigenpolytope Universality and Graphical Designs

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-03-05 DOI:10.1137/22m1528768

Catherine Babecki, David Shiroma

引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 947-964, March 2024.
Abstract. We show that the eigenpolytopes of graphs are universal in the sense that every polytope, up to affine equivalence, appears as the eigenpolytope of some positively weighted graph. We next extend the theory of graphical designs, which are quadrature rules for graphs, to positively weighted graphs. Through Gale duality for polytopes, we show a bijection between graphical designs and the faces of eigenpolytopes. This bijection proves the existence of graphical designs with positive quadrature weights and upper bounds the size of a minimal graphical design. Connecting this bijection with the universality of eigenpolytopes, we establish three complexity results: It is strongly NP-complete to determine if there is a graphical design smaller than the mentioned upper bound, it is NP-hard to find a smallest graphical design, and it is #P-complete to count the number of minimal graphical designs.

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特征多面体普遍性和图形设计

SIAM 离散数学杂志》，第 38 卷，第 1 期，第 947-964 页，2024 年 3 月。摘要我们证明了图的特征多面体是普适的，即每个多面体（直到仿射等价）都是某个正向加权图的特征多面体。接下来，我们将图形设计理论（即图形的正交规则）扩展到正加权图形。通过多面体的盖尔对偶性，我们展示了图形设计与特征多面体的面之间的双射关系。这个偏射证明了具有正二次权重的图形设计的存在，并给出了最小图形设计的大小上限。将这一偏射与特征多面体的普遍性联系起来，我们建立了三个复杂性结果：确定是否存在小于上述上限的图形设计是强 NP-完全的，找到最小图形设计是 NP-困难的，计算最小图形设计的数量是 #P- 完全的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution. Topics include but are not limited to: properties of and extremal problems for discrete structures combinatorial optimization, including approximation algorithms algebraic and enumerative combinatorics coding and information theory additive, analytic combinatorics and number theory combinatorial matrix theory and spectral graph theory design and analysis of algorithms for discrete structures discrete problems in computational complexity discrete and computational geometry discrete methods in computational biology, and bioinformatics probabilistic methods and randomized algorithms.