Arc Connectivity and Submodular Flows in Digraphs

IF 1 2区 数学 Q1 MATHEMATICS
Ahmad Abdi, Gérard Cornuéjols, Giacomo Zambelli
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引用次数: 0

Abstract

Let \(D=(V,A)\) be a digraph. For an integer \(k\ge 1\), a k-arc-connected flip is an arc subset of D such that after reversing the arcs in it the digraph becomes (strongly) k-arc-connected. The first main result of this paper introduces a sufficient condition for the existence of a k-arc-connected flip that is also a submodular flow for a crossing submodular function. More specifically, given some integer \(\tau \ge 1\), suppose \(d_A^+(U)+(\frac{\tau }{k}-1)d_A^-(U)\ge \tau \) for all \(U\subsetneq V, U\ne \emptyset \), where \(d_A^+(U)\) and \(d_A^-(U)\) denote the number of arcs in A leaving and entering U, respectively. Let \({\mathcal {C}}\) be a crossing family over ground set V, and let \(f:{\mathcal {C}}\rightarrow {\mathbb {Z}}\) be a crossing submodular function such that \(f(U)\ge \frac{k}{\tau }(d_A^+(U)-d_A^-(U))\) for all \(U\in {\mathcal {C}}\). Then D has a k-arc-connected flip J such that \(f(U)\ge d_J^+(U)-d_J^-(U)\) for all \(U\in {\mathcal {C}}\). The result has several applications to Graph Orientations and Combinatorial Optimization. In particular, it strengthens Nash-Williams’ so-called weak orientation theorem, and proves a weaker variant of Woodall’s conjecture on digraphs whose underlying undirected graph is \(\tau \)-edge-connected. The second main result of this paper is even more general. It introduces a sufficient condition for the existence of capacitated integral solutions to the intersection of two submodular flow systems. This sufficient condition implies the classic result of Edmonds and Giles on the box-total dual integrality of a submodular flow system. It also has the consequence that in a weakly connected digraph, the intersection of two submodular flow systems is totally dual integral.

Abstract Image

数图中的弧连接和次模流
让(D=(V,A)\)是一个数图。对于一个整数 \(k\ge 1\), k-弧连接的翻转是 D 的一个弧子集,使得在翻转其中的弧之后,数图变成(强)k-弧连接。本文的第一个主要结果介绍了一个充分条件,即对于一个交叉子模态函数来说,k-弧连接翻转也是一个子模态流。更具体地说,给定某个整数 \(\tau \ge 1\), 假设 \(d_A^+(U)+(\frac\tau }{k}-1)d_A^-(U)\ge \tau \) 对于所有 \(U\subsetneq V. U)都存在、其中 \(d_A^+(U)\) 和 \(d_A^-(U)\) 分别表示 A 中离开 U 和进入 U 的弧的数目。让 \({\mathcal {C}}\) 是地面集 V 上的一个交叉族,让 \(f:{\mathcal {C}}\rightarrow {\mathbb {Z}}\) 是一个交叉子模函数,使得 \(f(U)ge \frac{k}\{tau }(d_A^+(U)-d_A^-(U))\) for all \(Uin {\mathcal {C}}\).那么 D 有一个 k 弧连接的翻转 J,对于所有的 (U/in {mathcal {C}}\) 来说,(f(U)\ge d_J^+(U)-d_J^-(U)\) 是这样的。这个结果在图定向和组合优化中有一些应用。特别是,它加强了纳什-威廉姆斯(Nash-Williams)所谓的弱定向定理,并证明了伍德尔猜想(Woodall's conjecture)在底层无向图是(\tau \)边连接的数图上的一个较弱变体。本文的第二个主要结果更具一般性。它引入了两个子模流系统交集存在容积积分解的充分条件。这个充分条件意味着埃德蒙兹和贾尔斯关于子模态流动系统的盒总对偶积分性的经典结果。它的另一个结果是,在弱连接的数字图中,两个子模块流系统的交集是完全对偶积分的。
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
Combinatorica
Combinatorica 数学-数学
CiteScore
1.90
自引率
0.00%
发文量
45
审稿时长
>12 weeks
期刊介绍: COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are - Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups). - Combinatorial optimization. - Combinatorial aspects of geometry and number theory. - Algorithms in combinatorics and related fields. - Computational complexity theory. - Randomization and explicit construction in combinatorics and algorithms.
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