列表$H$的稀疏化下限-着色
Sparsification Lower Bounds for List $H$-Coloring
摘要：
我们调查列表$H$ -着色问题，图着色的推广，要求是否输入图$G$承认一个同态的无向图$H$（可能与循环），使每个顶点$v \in V(G)$被映射到一个顶点上的列表$L(v) \subseteq V(H)$。Feder，Hell，and Huang [JGT 2003]的一个重要结果表明，如果$H$是一个所谓的双弧图，则列表$H$ -着色是多项式时间可解的，否则是NP-完全的.我们从多项式时间稀疏化的角度研究了问题的NP完全情形：对于某些$\varepsilon > 0$，一个$n$顶点实例能否有效地简化为一个比特大小为$O(n^{2-\varepsilon})$的等价实例？我们证明，如果$H$是不是一个双弧图，那么列表$H$ -着色不承认这样的稀疏化算法，除非$NP \subseteq coNP/poly$。我们的证明相结合的技术，从核化下界的研究图$H$，这不是双弧图的结构。
英文摘要：
We investigate the List $H$-Coloring problem, the generalization of graph coloring that asks whether an input graph $G$ admits a homomorphism to the undirected graph $H$ (possibly with loops), such that each vertex $v \in V(G)$ is mapped to a vertex on its list $L(v) \subseteq V(H)$. An important result by Feder, Hell, and Huang [JGT 2003] states that List $H$-Coloring is polynomial-time solvable if $H$ is a so-called bi-arc graph, and NP-complete otherwise. We investigate the NP-complete cases of the problem from the perspective of polynomial-time sparsification: can an $n$-vertex instance be efficiently reduced to an equivalent instance of bitsize $O(n^{2-\varepsilon})$ for some $\varepsilon > 0$? We prove that if $H$ is not a bi-arc graph, then List $H$-Coloring does not admit such a sparsification algorithm unless $NP \subseteq coNP/poly$. Our proofs combine techniques from kernelization lower bounds with a study of the structure of graphs $H$ which are not bi-arc graphs.
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