arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2305.04767cs.CCmath.CO
收藏

关于CFI图的#P-困难性计数

Count on CFI graphs for #P-hardness

Radu Curticapean

备注：
20 pages
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

AI总结：

本文提出基于Cai-Fürer-Immerman图积的新归约技术，将彩色大树宽图同态计数问题在$4^{\Delta(S)}$时间内归约到无色版本，从而证明多个计数问题的#P-困难性。

摘要：

给定图$H$和$G$，可能带有顶点着色，同态是一个函数$f:V(H)\to V(G)$，它保持颜色和边。许多有趣的计数问题（例如子图和诱导子图计数）是同态计数的有限线性组合$p(\cdot)=\sum_{H}\alpha_{H}\hom(H,\cdot)$，并且已知这样的线性组合当且仅当它们包含一个大树宽图$S$时才难以评估。困难性可以通过两步证明：首先，对于彩色（即双射着色）大树宽图$S$，问题$\hom(S,\cdot)$被证明是困难的。在第二步中，这些问题被归约到包含$S$的无色版本$S^{\circ}$的同态计数的有限线性组合。这一步可以通过容斥原理在$2^{|E(S)|}\mathrm{poly}(n,s)$时间内执行，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。我们表明第二步甚至可以在$4^{\Delta(S)}\mathrm{poly}(n,s)$时间内执行，其中$\Delta(S)$是$S$的最大度。我们的归约基于与Cai-Fürer-Immerman图的图积，这是一种可能具有独立意义的新技术。对于最大度为常数的彩色图$S$，该技术产生从$\hom(S,\cdot)$到涉及$S^{\circ}$的同态计数线性组合的多项式时间归约。在某些条件下，实际上只要$S^{\circ}$的一个超图$T$包含在目标线性组合中就足够了。新的归约为几个计数问题产生了$\mathsf{\\#P}$-困难性结果，这些问题以前只能在参数化复杂性假设下研究。这包括在输入来自受限图类的图和一个一般图$G$时，计算从$H$到$G$的同态或（诱导）子图副本的数量。

英文摘要：

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 $p(\cdot)=\sum_{H}α_{H}\hom(H,\cdot)$ 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 $4^{Δ(S)}\mathrm{poly}(n,s)$, where $Δ(S)$ is the maximum degree of $S$. Our reduction is based on graph products with Cai-Fürer-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 $G$.

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。

↑