On characterizing the extreme points of the generalized transitive tournament polytope

Q2 Mathematics

Electronic Notes in Discrete Mathematics Pub Date : 2018-07-01 DOI:10.1016/j.endm.2018.06.047

Konstantinos Papalamprou

引用次数: 0

Abstract

A non-negative $n \times n$ matrix [ $x_{i j}$ ] is called generalized tournament, denoted GTT(n), if: $x_{i i} = 0$ (for all i), $x_{i j} + x_{j i} = 1$ (for all(i,j) with $i \neq j$ ) and $1 \leq x_{i j} + x_{j k} + x_{k i} \leq 2$ (for all (i,j,k) with $i, j, k$ pairwise distinct). In [9], using hypergraphs associated with GTT matrices, it has been shown that for $n \leq 6$ all the vertices of the GTT(n) polytope are half-integral. In this work, we show that these matrices belong to the class of 2-regular matrices and highlight the related optimization implications. Finally, based on our approach and known partial results, conjectures on characterizing the extreme points of the GTT(n) polytope for $n \geq 7$ are provided.

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关于广义可传递竞赛多面体极值点的刻画

一个非负的n×n矩阵[xij]被称为广义竞赛，记作GTT(n)，如果:xii=0(对于所有i)， xij+xji=1(对于所有(i,j)且i≠j)， 1≤xij+xjk+xki≤2(对于所有(i,j,k)且i,j,k成对不同)。在[9]中，利用与GTT矩阵相关的超图，证明了当n≤6时，GTT(n)多面体的所有顶点都是半积分。在这项工作中，我们证明了这些矩阵属于2正则矩阵类，并强调了相关的优化含义。最后，根据我们的方法和已知的部分结果，给出了n≥7时GTT(n)多面体极值点表征的猜想。

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

Electronic Notes in Discrete Mathematics Mathematics-Discrete Mathematics and Combinatorics

CiteScore

1.30

自引率

0.00%

发文量

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