From Graph Properties to Graph Parameters: Tight Bounds for Counting on Small Subgraphs

TL;DR

This paper establishes tight lower bounds for counting induced subgraphs with certain properties, extending previous results to more general graph parameters and demonstrating the computational hardness under ETH.

Contribution

It generalizes lower bounds for counting problems to a wider class of graph parameters, including non-binary and infinite codomain functions, and extends hardness results to modular and multicolored variants.

Findings

Lower bounds hold for all nontrivial edge-monotone parameters with finite codomain.

Assuming ETH, no subexponential time algorithm exists for these problems.

The bounds also apply to modular counting and multicolored versions.

Abstract

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.