在键多面体上

IF 0.5 4区 数学 Q3 MATHEMATICS
Markus Chimani, Martina Juhnke-Kubitzke, Alexander Nover
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引用次数: 0

摘要

摘要文献中对最大切问题及其对应的多面体进行了大量的研究,但对自然密切相关的变异体最大键的研究却很少。在此,给定图G = (V, E),在G [S]和G [V \ S]都连通的条件下,求出S与S的最大切量δ(S)。观察到,最大切割和最大键都可以被视为传统最小切割的逆问题,因为在那里,连通性自然出现在最优解中。键多面体是键的所有入射向量的凸包。类似于相应的连接优化问题,键合多面体与切割多面体密切相关。虽然后者已被深入研究,但尚无关于键多面体的结果。我们开始对后者进行结构性研究,这也使我们能够推断出算法的结果。我们研究了切割和键多面体之间的关系,以及在解决方案中需要连通性时出现的额外复杂性。我们研究了图修饰对键多面体及其切面的影响,类似于Barahona, Grötschel和Mahjoub [4;[3]德撒和劳伦特[17];15;16)。此外,我们还研究了键多面体由边和环引起的面定义不等式。特别是,这些得到了环的键多面体和3连通平面(k5−e)-小自由图的完整线性描述。最后,我们将任意图上的最大键问题简化为3连通图上的最大键问题。这产生了(k5−e)次自由图上最大键的线性时间算法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the bond polytope
Abstract While the maximum cut problem and its corresponding polytope has received a lot of attention inliterature, comparably little is known about the natural closely related variant maximum bond. Here, given a graph G = (V, E) , we ask for a maximum cut δ(S) ⊆ E with S ⊆ V under the restriction that both G [ S ] as well as G [ V \ S ] are connected. Observe that both the maximum cut and the maximum bond can be seen as inverse problems to the traditional minimum cut, as there, the connectivity arises naturally in optimal solutions. The bond polytope is the convex hull of all incidence vectors of bonds. Similar to the connection of the corresponding optimization problems, the bond polytope is closely related to the cut polytope. While the latter has been intensively studied, there are no results on bond polytopes. We start a structural study of the latter, which additionally allows us to deduce algorithmic consequences. We investigate the relation between cut- and bond polytopes and the additional intricacies that arise when requiring connectivity in the solutions. We study the effect of graph modifications on bond polytopes and their facets, akin to what has been spearheaded for cut polytopes by Barahona, Grötschel and Mahjoub [4; 3] and Deza and Laurant [17; 15; 16]. Moreover, we study facet-defining inequalities arising from edges and cycles for bond polytopes. In particular, these yield a complete linear description of bond polytopes of cycles and 3-connected planar ( K 5 − e )-minor free graphs. Finally, we present a reduction of the maximum bond problem on arbitrary graphs to the maximum bond problem on 3-connected graphs. This yields a linear time algorithm for maximum bond on ( K 5 − e )-minor free graphs.
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来源期刊
Advances in Geometry
Advances in Geometry 数学-数学
CiteScore
1.00
自引率
0.00%
发文量
31
审稿时长
>12 weeks
期刊介绍: Advances in Geometry is a mathematical journal for the publication of original research articles of excellent quality in the area of geometry. Geometry is a field of long standing-tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity; geometric ideas and geometric language permeate all of mathematics.
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