We study homomorphism polynomials, which are polynomials that enumerate all homomorphisms from a pattern graph $H$ to $n$-vertex graphs. These polynomials have received a lot of attention recently for their crucial role in several new algorithms for counting and detecting graph patterns, and also for obtaining natural polynomial families which are complete for algebraic complexity classes $\mathsf{VBP}$, $\mathsf{VP}$, and $\mathsf{VNP}$. We discover that, in the monotone setting, the formula complexity, the ABP complexity, and the circuit complexity of such polynomial families are exactly characterized by the treedepth, the pathwidth, and the treewidth of the pattern graph respectively. Furthermore, we establish a single, unified framework, using our characterization, to collect several known results that were obtained independently via different methods. For instance, we attain superpolynomial separations between circuits, ABPs, and formulas in the monotone setting, where the polynomial families separating the classes all correspond to well-studied combinatorial problems. Moreover, our proofs rediscover fine-grained separations between these models for constant-degree polynomials. The characterization additionally yields new space-time efficient algorithms for several pattern detection and counting problems.

中文翻译：

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图同态多项式：算法和复杂性

我们研究同态多项式，它们是枚举从模式图 $H$ 到 $n$-顶点图的所有同态的多项式。这些多项式最近受到了很多关注，因为它们在几种用于计数和检测图模式的新算法中的关键作用，以及获得对于代数复杂性类 $\mathsf{VBP}$, $\mathsf{ 是完整的自然多项式族的关键作用VP}$ 和 $\mathsf{VNP}$。我们发现，在单调设置中，这些多项式族的公式复杂度、ABP复杂度和电路复杂度分别由模式图的树深、路径宽度和树宽精确表征。此外，我们建立了一个单一的、统一的框架，使用我们的特征，收集通过不同方法独立获得的几个已知结果。例如，我们在单调设置中实现了电路、ABP 和公式之间的超多项式分离，其中分离类的多项式族都对应于经过充分研究的组合问题。此外，我们的证明重新发现了这些模型之间的常项多项式的细粒度分离。该表征还为多个模式检测和计数问题产生了新的时空高效算法。我们的证明重新发现了这些模型之间的常次多项式的细粒度分离。该表征还为多个模式检测和计数问题产生了新的时空高效算法。我们的证明重新发现了这些模型之间的常次多项式的细粒度分离。该表征还为多个模式检测和计数问题产生了新的时空高效算法。