A graph property is a function $\Phi$ that maps every graph to {0, 1} and is invariant under isomorphism. In the $\#IndSub(\Phi)$ problem, given a graph $G$ and an integer $k$, the task is to count the number of $k$-vertex induced subgraphs $G'$ with $\Phi(G')=1$. $\#IndSub(\Phi)$ can be naturally generalized to graph parameters, that is, to functions $\Phi$ on graphs that do not necessarily map to {0, 1}: now the task is to compute the sum $\sum_{G'} \Phi(G')$ taken over all $k$-vertex induced subgraphs $G'$. This problem setting can express a wider range of counting problems (for instance, counting $k$-cycles or $k$-matchings) and can model problems involving expected values (for instance, the expected number of components in a subgraph induced by $k$ random vertices). Our main results are lower bounds on $\#IndSub(\Phi)$ in this setting, which simplify, generalize, and tighten the recent lower bounds of D\"oring, Marx, and Wellnitz [STOC'24] in various ways. (1) We show a lower bound for every nontrivial edge-monotone graph parameter $\Phi$ with finite codomain (not only for parameters that take value in {0, 1}). (2) The lower bound is tight: we show that, assuming ETH, there is no $f(k)n^{o(k)}$ time algorithm. (3) The lower bound applies also to the modular counting versions of the problem. (4) The lower bound applies also to the multicolored version of the problem. We can extend the #W[1]-hardness result to the case when the codomain of $\Phi$ is not finite, but has size at most $(1 - \varepsilon)\sqrt{k}$ on $k$-vertex graphs. However, if there is no bound on the size of the codomain, the situation changes significantly: for example, there is a nontrivial edge-monotone function $\Phi$ where the size of the codomain is $k$ on $k$-vertex graphs and $\#IndSub(\Phi)$ is FPT.

翻译：图性质是一个将每个图映射到{0, 1}且在同构下保持不变的函数Φ。在#IndSub(Φ)问题中，给定图G和整数k，任务是计算满足Φ(G')=1的k顶点诱导子图G'的数量。#IndSub(Φ)可以自然地推广到图参数，即推广到不一定映射到{0, 1}的图函数Φ：此时任务是计算所有k顶点诱导子图G'上的和∑Φ(G')。此问题框架能表达更广泛的计数问题（例如计算k环或k匹配），并能建模涉及期望值的问题（例如由k个随机顶点诱导的子图中连通分量的期望数量）。我们的主要成果是在此框架下#IndSub(Φ)的下界，这些下界从多个方面简化、推广并强化了Döring、Marx和Wellnitz [STOC'24]的最新下界。(1) 我们证明了每个具有有限值域的非平凡边单调图参数Φ的下界（不仅限于取值在{0, 1}的参数）。(2) 该下界是紧的：我们证明在ETH假设下，不存在f(k)n^{o(k)}时间算法。(3) 该下界同样适用于问题的模计数版本。(4) 该下界同样适用于问题的多色版本。我们可以将#W[1]难性结果扩展到Φ的值域非有限但在k顶点图上规模不超过(1-ε)√k的情形。然而，若值域规模无界，情况将发生显著变化：例如存在非平凡边单调函数Φ，其在k顶点图上的值域规模为k，而#IndSub(Φ)是FPT。