arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2403.07597math.CO
收藏

半完全多部有向图中的广义路径与圈

Generalized paths and cycles in semicomplete multipartite digraphs

Jørgen Bang-Jensen, Yun Wang, Anders Yeo

阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

AI总结：

研究半完全多部有向图中广义路径与圈的结构和算法性质，识别出可多项式时间求解或近似求解的0-1加权有向TSP实例类，同时证明两个相关圈问题为NP完全。

摘要：

众所周知且易于证明，即便有向旅行商问题的下述版本也是NP完全的：给定强连通完全有向图$D=(V,A)$、代价函数$w: A\rightarrow \{0,1\}$以及自然数$K$，判断$D$中是否存在代价不超过$K$的有向哈密顿圈。我们针对$\{0,1\n$加权的半完全有向图研究该问题的一个变体，其中代价为1的弧集构成一组顶点不相交的完全有向图。若一个有向图可通过以下方式从半完全有向图$D$得到：选取$V(D)$的一组顶点不相交子集$X_1,\ldots,X_c$，随后删除所有两个端点均位于某个$X_i$内部的弧，则称该有向图为**半完全多部有向图（semicomplete multipartite digraph）**。设$D$为具有上述代价函数$w$的半完全有向图，其中当且仅当弧$a$属于子集$X_1,\ldots,X_c$中某一个的内部时$w(a)=1$，并设$D^*$为我们通过删除所有$X_i$内部的弧得到的对应半完全多部有向图（\smd{}）。那么$D$中的每个圈$C$都对应$D^*$中的一个**广义圈**$C^g$：若$w(C)=0$，则广义圈就是圈$C$本身；否则，它是我们通过删除$C$上所有代价为1的弧得到的两条或更多条路径的集合。类似地，我们可以定义半完全多部有向图中的**广义路径**$P^g$。本文的目的是研究半完全多部有向图中广义路径与圈的结构性质与算法性质。这使得我们能够识别出可在多项式时间内求解的有向$\{0,1\n$加权TSP实例类别，以及其他可在多项式时间内得到非常接近最优解的实例类别。结合这些结果，我们还证明了关于半完全多部有向图中与所有部集相交的圈的性质的两个自然问题是NP完全的。

英文摘要：

It is well-known and easy to show that even the following version of the directed travelling salesman problem is NP-complete: Given a strongly connected complete digraph $D=(V,A)$, a cost function $w: A\rightarrow \{0,1\}$ and a natural number $K$; decide whether $D$ has a directed Hamiltonian cycle of cost at most $K$. We study the following variant of this problem for $\{0,1\}$-weighted semicomplete digraphs where the set of arcs which have cost 1 form a collection of vertex-disjoint complete digraphs. A digraph is \textbf{semicomplete multipartite} if it can be obtained from a semicomplete digraph $D$ by choosing a collection of vertex-disjoint subsets $X_1,\ldots{},X_c$ of $V(D)$ and then deleting all arcs both of whose end-vertices lie inside some $X_i$. Let $D$ be a semicomplete digraph with a cost function $w$ as above, where $w(a)=1$ precisely when $a$ is an arc inside one of the subsets $X_1,\ldots{},X_c$ and let $D^*$ be the corresponding \smd{} that we obtain by deleting all arcs inside the $X_i$'s. Then every cycle $C$ of $D$ corresponds to a {\bf generalized cycle} $C^g$ of $D^*$ which is either the cycle $C$ itself if $w(C)=0$ or a collection of two or more paths that we obtain by deleting all arcs of cost 1 on $C$. Similarly we can define a {\bf generalized path} $P^g$ in a semicomplete multipartite digraph. The purpose of this paper is to study structural and algorithmic properties of generalized paths and cycles in semicomplete multipartite digraphs. This allows us to identify classes of directed $\{0,1\}$-weighted TSP instances that can be solved in polynomial time as well as others for which we can get very close to the optimum in polynomial time. Along with these results we also show that two natural questions about properties of cycles meeting all partite sets in semicomplete multipartite digraphs are NP-complete.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑