arXiv 1902.022012019-02cs.DS

关于$H$ -染色逼近的一个二分法

Toward a Dichotomy for Approximation of $H$-coloring

Akbar Rafiey, Arash Rafiey, Thiago Santos

摘要：

给定两个（di）图G、H和一个代价函数$c:V(G)\times V(H) \to \mathbb{Q}_{\geq 0}\cup\{+\infty\}$，在最小代价同态问题MinHOM（H）中，目标是找到一个最小化$\sum\limits_{v\in V(G)}c(v,f(v))$的同态$f:V(G)\to V(H)$（也称为H-着色）。这个问题的精确最小化的复杂性是很好理解的[34]，并且MinHOM（H）是多项式时间可解的有向图类H是所有有向图的一个小子集。在本文中，我们考虑的MinHOM在一个常数因子的近似。对于有向图，如果H包含有向图星状三元组（DAT），则MinHOM（H）是不可逼近的。我们向近似情况的二分法分类迈出了重要的一步。当H是一个图时，我们给出了MinHOM（H）的一个二分法.对于有向图，我们提供了两类重要的有向图的常数因子近似算法，即双弧有向图（具有保守半格多态性或最小序的有向图）和k弧有向图（具有扩展最小序的有向图）。具体来说，我们表明：1。图的二分法：证明了MinHOM（H）有一个$2|V(H)|$ -近似算法，如果H是保守多数多态图（即H是一个双弧图），否则它是不可近似的; 2.当H是双弧有向图时，MinHOM（H）有一个$|V(H)|^2$ -近似算法; 3.如果H是k-弧有向图，则MinHOM（H）有一个$|V(H)|^2$ -近似算法.总之，我们显示了这些结果的重要性，并提供了见解，实现二分法分类近似的情况下。我们的常数因子取决于H的大小。然而，我们的算法的实现提供了一个更好的近似比。它留下了开放的研究分类的有向图H，其中MinHOM（H）允许一个常数因子近似算法，是独立于H。

英文摘要：

Given two (di)graphs G, H and a cost function $c:V(G)\times V(H) \to \mathbb{Q}_{\geq 0}\cup\{+\infty\}$, in the minimum cost homomorphism problem, MinHOM(H), goal is finding a homomorphism $f:V(G)\to V(H)$ (a.k.a H-coloring) that minimizes $\sum\limits_{v\in V(G)}c(v,f(v))$. The complexity of exact minimization of this problem is well understood [34], and the class of digraphs H, for which the MinHOM(H) is polynomial time solvable is a small subset of all digraphs. In this paper, we consider the approximation of MinHOM within a constant factor. For digraphs, MinHOM(H) is not approximable if H contains a digraph asteroidal triple (DAT). We take a major step toward a dichotomy classification of approximable cases. We give a dichotomy classification for approximating the MinHOM(H) when H is a graph. For digraphs, we provide constant factor approximation algorithms for two important classes of digraphs, namely bi-arc digraphs (digraphs with a conservative semi-lattice polymorphism or min-ordering), and k-arc digraphs (digraphs with an extended min-ordering). Specifically, we show that: 1. Dichotomy for Graphs: MinHOM(H) has a $2|V(H)|$-approximation algorithm if graph H admits a conservative majority polymorphims (i.e. H is a bi-arc graph), otherwise, it is inapproximable; 2. MinHOM(H) has a $|V(H)|^2$-approximation algorithm if H is a bi-arc digraph; 3. MinHOM(H) has a $|V(H)|^2$-approximation algorithm if H is a k-arc digraph. In conclusion, we show the importance of these results and provide insights for achieving a dichotomy classification of approximable cases. Our constant factors depend on the size of H. However, the implementation of our algorithms provides a much better approximation ratio. It leaves open to investigate a classification of digraphs H, where MinHOM(H) admits a constant factor approximation algorithm that is independent of H.

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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