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arXiv cs/0507017cs.DM
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半完全有向图的最小代价和列表同态

Minimum Cost and List Homomorphisms to Semicomplete Digraphs

G. Gutin, A. Rafiey, A. Yeo

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摘要：

下面的优化问题是在\cite{gutinDAM}中引入的，它是由国防后勤中的一个现实问题激发的。假设我们给定一对有向图$D,H$和一个正代价$c_i(u)$，每个$u\in V(D)$和$i\in V(H)$。$D$到$H$的同态$f$的代价是$\sum_{u\in V(D)}c_{f(u)}(u)$。对于固定有向图$H$，$H$的最小代价同态问题MinHOMP（$H$）陈述如下：对于输入有向图$D$和每个$u\in V(D)$和$i\in V(H)$的代价$c_i(u)$，验证是否存在$D$到$H$的同态，并且如果存在，找到这样的最小代价同态。当$H$是半完全有向图时，得到了列表同态问题和MinHOMP（$H$）计算复杂性的二分分类.当$H$是一个半完全有向图时，我们对列表同态问题的二分法与Bang-Jensen，Hell和MacGillivray在1988年对同态问题的二分法一致：如果$H$至多有一个圈，则这两个问题都是多项式可解的;否则，这两个问题都是NP-完全的. \MiP的二分法是不同的：如果$H$是无环的或者$H$是长度为2或3的环，则问题是多项式时间可解的;否则，问题是NP-难的。

英文摘要：

The following optimization problem was introduced in \cite{gutinDAM}, where it was motivated by a real-world problem in defence logistics. Suppose we are given a pair of digraphs $D,H$ and a positive cost $c_i(u)$ for each $u\in V(D)$ and $i\in V(H)$. The cost of a homomorphism $f$ of $D$ to $H$ is $\sum_{u\in V(D)}c_{f(u)}(u)$. For a fixed digraph $H$, the minimum cost homomorphism problem for $H$, MinHOMP($H$), is stated as follows: For an input digraph $D$ and costs $c_i(u)$ for each $u\in V(D)$ and $i\in V(H)$, verify whether there is a homomorphism of $D$ to $H$ and, if it exists, find such a homomorphism of minimum cost. We obtain dichotomy classifications of the computational complexity of the list homomorphism problem and MinHOMP($H$), when $H$ is a semicomplete digraph (a digraph in which every two vertices have at least one arc between them). Our dichotomy for the list homomorphism problem coincides with the one obtained by Bang-Jensen, Hell and MacGillivray in 1988 for the homomorphism problem when $H$ is a semicomplete digraph: both problems are polynomial solvable if $H$ has at most one cycle; otherwise, both problems are NP-complete. The dichotomy for \MiP is different: the problem is polynomial time solvable if $H$ is acyclic or $H$ is a cycle of length 2 or 3; otherwise, the problem is NP-hard.

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。

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