arXiv 0712.08042007-12-05cs.DM

局部半完全和拟传递有向图的最小代价同态

Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs

A. Gupta, G. Gutin, M. Karimi, E. J. Kim, A. Rafiey

摘要：

对于有向图$G$和$H$，$G$到$H$的同态是映射$f:\ V(G)\dom V(H)$，使得$uv\in A(G)$蕴涵$f(u)f(v)\in A(H)$。此外，如果每个顶点$u \in V(G)$与成本$c_i(u), i \in V(H)$相关联，则同态$f$的成本是$\sum_{u\in V(G)}c_{f(u)}(u)$。对于每个固定的有向图$H$，$H$的最小代价同态问题，记为MinHOM（$H$），可以公式化如下：给定一个输入有向图$G$，连同代价$c_i(u)$，$u\in V(G)$，$i\in V(H)$，判断是否存在$G$到$H$的同态，如果存在，找出一个最小代价。最小代价同态问题包含（或涉及）许多研究得很好的优化问题，如最小代价色划分和修复分析问题。本文主要研究了竞赛图的两个著名推广--局部半完全有向图和拟传递有向图的最小费用同态问题。使用图论的两个有向图类的特征结果，我们得到了一个完整的二分法分类的复杂性的最小成本同态问题的两个类。

英文摘要：

For digraphs $G$ and $H$, a homomorphism of $G$ to $H$ is a mapping $f:\ V(G)\dom V(H)$ such that $uv\in A(G)$ implies $f(u)f(v)\in A(H)$. If, moreover, each vertex $u \in V(G)$ is associated with costs $c_i(u), i \in V(H)$, then the cost of a homomorphism $f$ is $\sum_{u\in V(G)}c_{f(u)}(u)$. For each fixed digraph $H$, the minimum cost homomorphism problem for $H$, denoted MinHOM($H$), can be formulated as follows: Given an input digraph $G$, together with costs $c_i(u)$, $u\in V(G)$, $i\in V(H)$, decide whether there exists a homomorphism of $G$ to $H$ and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems such as the minimum cost chromatic partition and repair analysis problems. We focus on the minimum cost homomorphism problem for locally semicomplete digraphs and quasi-transitive digraphs which are two well-known generalizations of tournaments. Using graph-theoretic characterization results for the two digraph classes, we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for both classes.

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。

摘要：

英文摘要：

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