图同态问题的下界
Lower Bounds for the Graph Homomorphism Problem
摘要：
图同态问题（HOM）询问给定$n$ -顶点图$G$的顶点是否可以映射到给定$h$ -顶点图$H$的顶点，使得$G$的每条边映射到$H$的一条边。该问题推广了图着色问题，同时也可以看作是$2$-CSP问题的一个特例。本文在指数时间假设（ETH）下证明了HOM的几个下界。主要结果是一个下界$2^{Ω\left( \frac{n \log h}{\log \log h}\right)}$。这排除了单指数算法的存在性，并表明平凡上界$2^{{\mathcal O}(n\log{h})}$几乎是渐近紧的。我们还研究了图$G$和$H$的哪些性质使得HOM $(G,H)$难以求解。容易观察到的是，${\mathcal O}(h^n)$上界可以改进为${\mathcal O}(h^{\operatorname{vc}(G)})$，其中$\operatorname{vc}(G)$是$G$的顶点覆盖的最小尺寸。第二个下界$h^{Ω(\operatorname{vc}(G))}$表明上界是渐近紧的。关于“右手边”图$H$的性质，已知HOM $(G,H)$可以在时间$(f(Δ(H)))^n$和$(f(\operatorname{tw}(H)))^n$中求解，其中$Δ(H)$是$H$的最大度，$\operatorname{tw}(H)$是$H$的树宽。这给出了有界最大度或有界树宽的图的单指数算法。由于色数$χ(H)$不超过$\operatorname{tw}(H)$和$Δ(H)+1$，自然会问是否可以获得关于$χ(H)$的类似上界。我们提供了一个否定的答案，这个问题建立一个下限$(f(χ(H)))^n$的任何功能$f$。我们还观察到，类似的下界可以获得局部内射同态。
英文摘要：
The graph homomorphism problem (HOM) asks whether the vertices of a given $n$-vertex graph $G$ can be mapped to the vertices of a given $h$-vertex graph $H$ such that each edge of $G$ is mapped to an edge of $H$. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the $2$-CSP problem. In this paper, we prove several lower bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main result is a lower bound $2^{Ω\left( \frac{n \log h}{\log \log h}\right)}$. This rules out the existence of a single-exponential algorithm and shows that the trivial upper bound $2^{{\mathcal O}(n\log{h})}$ is almost asymptotically tight. We also investigate what properties of graphs $G$ and $H$ make it difficult to solve HOM$(G,H)$. An easy observation is that an ${\mathcal O}(h^n)$ upper bound can be improved to ${\mathcal O}(h^{\operatorname{vc}(G)})$ where $\operatorname{vc}(G)$ is the minimum size of a vertex cover of $G$. The second lower bound $h^{Ω(\operatorname{vc}(G))}$ shows that the upper bound is asymptotically tight. As to the properties of the "right-hand side" graph $H$, it is known that HOM$(G,H)$ can be solved in time $(f(Δ(H)))^n$ and $(f(\operatorname{tw}(H)))^n$ where $Δ(H)$ is the maximum degree of $H$ and $\operatorname{tw}(H)$ is the treewidth of $H$. This gives single-exponential algorithms for graphs of bounded maximum degree or bounded treewidth. Since the chromatic number $χ(H)$ does not exceed $\operatorname{tw}(H)$ and $Δ(H)+1$, it is natural to ask whether similar upper bounds with respect to $χ(H)$ can be obtained. We provide a negative answer to this question by establishing a lower bound $(f(χ(H)))^n$ for any function $f$. We also observe that similar lower bounds can be obtained for locally injective homomorphisms.
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