From Graph Properties to Graph Parameters: Tight Bounds for Counting on Small Subgraphs
Simon Döring, Dániel Marx, Philip Wellnitz
TL;DR
The paper broadens the study of counting small subgraphs by treating graph parameters Φ and analyzing the complexity of #IndSub(Φ) and its colored version. It develops a robust reduction framework using alternating enumerators, fixed-point analysis of p-subgroups, and inhabited graphs to transfer #Clique hardness to IndSub for nontrivial edge-monotone Φ with finite codomain, yielding tight ETH lower bounds depending on k and the growth parameter ψ. It extends the theory to modular counting and colored variants, showing that modulo counts remain hard and establishing when such problems admit fixed-parameter tractable solutions via the sub-basis representation and bounded-VC structure. The Sub-Basis viewpoint reveals when IndSub(Φ) can be efficiently computed as a linear combination of Sub(H→G) terms, and clarifies why cp-IndSub(Φ) can be easier in some unifying cases. Overall, the work delineates precise boundaries between tractable and intractable counting problems for small subgraphs under a broad class of graph parameters, under ETH/rETH assumptions, and highlights nuanced behavior when the codomain is unbounded or when modular counting is considered.
Abstract
A graph property is a function $Φ$ that maps every graph to {0, 1} and is invariant under isomorphism. In the $\#IndSub(Φ)$ problem, given a graph $G$ and an integer $k$, the task is to count the number of $k$-vertex induced subgraphs $G'$ with $Φ(G')=1$. $\#IndSub(Φ)$ can be naturally generalized to graph parameters, that is, to functions $Φ$ on graphs that do not necessarily map to {0, 1}: now the task is to compute the sum $\sum_{G'} Φ(G')$ 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 $\#IndSub(Φ)$ in this setting, which simplify, generalize, and tighten the recent lower bounds of Döring, Marx, and Wellnitz [STOC'24] in various ways. (1) We show a lower bound for every nontrivial edge-monotone graph parameter $Φ$ 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 $Φ$ 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 $Φ$ where the size of the codomain is $k$ on $k$-vertex graphs and $\#IndSub(Φ)$ is FPT.