AI 中文总结
研究图同态问题HOM在目标图H退化性方面的复杂性，通过对比其他图参数，发现其对HOM复杂性影响有差异，给出相关算法时间下限等结果，并引入无压缩障碍解释稀疏2-CSP下界不紧及多项式目标退化性更强下界难得出的原因。
AI 中文摘要
图同态问题HOM是：给定一个n个顶点的源图G和一个h个顶点的目标图H，是否存在从V(G)到V(H)的边保持映射？一种直接的暴力算法运行时间为O(2^{n log h})，并且已知在指数时间假设（ETH）下，不存在2^{o(n log h)}的算法。近年来，已确定了限制较少的图参数p，可在时间p(H)^{O(n)}内解决HOM。例如树宽、最大度和轨道数。另一方面，已知色数参数太小：在ETH下，HOM不能在时间χ(H)^{O(n)}内解决。我们从H的退化性角度研究HOM的复杂性。这可能是已知算法和硬度机制之间最自然未解决的图参数：一方面，有界树宽、有界最大度和有界轨道数都意味着有界退化性；另一方面，有界退化性意味着有界色数。我们的结果表明，同时，H的退化性对HOM复杂性的影响与先前研究的参数有显著差异。我们表明，在ETH下，对于任何作为n的函数的degen(H)值，都不存在2^{o(degen(H) n)}的算法。我们还表明，仅有限制的退化性并不能使目标大小良性：即使degen(H)至多为2且具有拟多项式大小的目标也会导致n^{Ω(n)}规模的硬度。最后，我们引入了一个无压缩障碍，解释了为什么在ETH下稀疏2-CSP的已知细粒度下界不紧。此外，它表明从稀疏3-SAT的标准归约不太可能得出多项式目标退化性更强的下界。
英文摘要
The graph homomorphism problem HOM is: given an $n$-vertex source graph $G$ and an $h$-vertex target graph $H$, is there a mapping from $V(G)$ to $V(H)$ that preserves edges? A straightforward brute-force algorithm for HOM has running time $O(2^{n \log h})$ and it is known that, under ETH, there are no $2^{o(n \log h)}$ algorithms. In recent years, less restrictive graph parameters $p$ have been identified that allow one to solve HOM in time $p(H)^{O(n)}$. Examples include treewidth, maximum degree, and track number. On the other hand, it is known that the chromatic number parameter is too small: under ETH, HOM cannot be solved in time $χ(H)^{O(n)}$. We study the complexity of HOM in terms of the degeneracy of $H$. This is perhaps the most natural unresolved graph parameter between the known algorithmic and hardness regimes: on the one hand, each of bounded treewidth, bounded maximum degree, and bounded track number implies bounded degeneracy; on the other hand, bounded degeneracy implies bounded chromatic number. Our results show that, at the same time, the influence of degeneracy of $H$ on the complexity of HOM differs significantly from that of the previously studied parameters. We show that, under ETH, there is no $2^{o(degen(H) n)}$ algorithm for any value of $degen(H)$ as a function of $n$. We also show that bounded degeneracy alone does not make target size benign: even targets with $degen(H)$ at most $2$ and quasi-polynomial size force $n^{Ω(n)}$-scale hardness. Finally, we introduce a no-compression barrier that explains why the known fine-grained lower bounds for sparse $2$-CSP are not tight under ETH. Moreover, it shows that substantially stronger lower bounds for polynomial-target degeneracy are unlikely to follow from standard reductions from sparse $3$-SAT.
