Given graphs $H$ and $G$, possibly with vertex-colors, a homomorphism is a function $f:V(H)\to V(G)$ that preserves colors and edges. Many interesting counting problems (e.g., subgraph and induced subgraph counts) are finite linear combinations $p(\cdot)=\sum_{H}\alpha_{H}\hom(H,\cdot)$ of homomorphism counts, and such linear combinations are known to be hard to evaluate iff they contain a large-treewidth graph $S$. The hardness can be shown in two steps: First, the problems $\hom(S,\cdot)$ for colorful (i.e., bijectively colored) large-treewidth graphs $S$ are shown to be hard. In a second step, these problems are reduced to finite linear combinations of homomorphism counts that contain the uncolored version $S^{\circ}$ of $S$. This step can be performed via inclusion-exclusion in $2^{|E(S)|}\mathrm{poly}(n,s)$ time, where $n$ is the size of the input graph and $s$ is the maximum number of vertices among all graphs in the linear combination. We show that the second step can be performed even in time $4^{\Delta(S)}\mathrm{poly}(n,s)$, where $\Delta(S)$ is the maximum degree of $S$. Our reduction is based on graph products with Cai-F\"urer-Immerman graphs, a novel technique that is likely of independent interest. For colorful graphs $S$ of constant maximum degree, this technique yields a polynomial-time reduction from $\hom(S,\cdot)$ to linear combinations of homomorphism counts involving $S^{\circ}$. Under certain conditions, it actually suffices that a supergraph $T$ of $S^{\circ}$ is contained in the target linear combination. The new reduction yields $\mathsf{\#P}$-hardness results for several counting problems that could previously be studied only under parameterized complexity assumptions. This includes the problems of counting, on input a graph from a restricted graph class and a general graph $G$, the homomorphisms or (induced) subgraph copies from $H$ in $G$.

翻译：给定图 $H$ 和 $G$（可能带有顶点颜色），同态是一个保持颜色和边的函数 $f:V(H)\to V(G)$。许多有趣的计数问题（例如子图计数和导出子图计数）是同态计数 $\hom(H,\cdot)$ 的有限线性组合 $p(\cdot)=\sum_{H}\alpha_{H}\hom(H,\cdot)$，且已知此类线性组合的计算困难性等价于其中是否包含一个大树宽图 $S$。该困难性可通过两步证明：首先，证明对于彩色（即双射着色）的大树宽图 $S$，问题 $\hom(S,\cdot)$ 是困难的；其次，将这些问题归约为包含 $S$ 的无色版本 $S^{\circ}$ 的同态计数有限线性组合。该步骤可通过容斥原理在 $2^{|E(S)|}\mathrm{poly}(n,s)$ 时间内完成，其中 $n$ 是输入图的大小，$s$ 是线性组合中所有图的最大顶点数。我们证明第二步甚至可以在 $4^{\Delta(S)}\mathrm{poly}(n,s)$ 时间内完成，其中 $\Delta(S)$ 是 $S$ 的最大度数。我们的归约基于图乘积与 Cai-F¨urer-Immerman 图，这是一项可能具有独立意义的新技术。对于最大度数为常数的彩色图 $S$，该技术可在多项式时间内将 $\hom(S,\cdot)$ 归约为包含 $S^{\circ}$ 的同态计数线性组合。在某些条件下，实际上只需目标线性组合中包含 $S^{\circ}$ 的一个超图 $T$ 即可。这一新归约为多个计数问题提供了 $\mathsf{\#P}$ 困难性结果，而此前这些问题只能在参数化复杂度假设下进行研究，包括：在输入图为受限图类、而 $G$ 为一般图的情况下，计数 $H$ 到 $G$ 的同态或（导出）子图副本数的问题。