Exact and Approximate Pattern Counting in Degenerate Graphs: New Algorithms, Hardness Results, and Complexity Dichotomies

TL;DR

This paper provides a comprehensive complexity classification for counting subgraphs and induced subgraphs in degenerate graphs, establishing both exact algorithms and hardness results, along with new approximate counting methods.

Contribution

It introduces explicit complexity classifications for counting problems in degenerate graphs, including tight bounds and algorithms, and extends to approximate counting techniques.

Findings

Counting subgraphs can be done in time depending on the largest induced matching or independence number.

Several pattern counting problems are shown to be hard under ETH, including k-matchings and k-paths.

New algorithms for approximate counting in degenerate graphs are proposed.

Abstract

We study the problems of counting the homomorphisms, counting the copies, and counting the induced copies of a $k$-vertex graph $H$ in a $d$-degenerate $n$-vertex graph $G$. Our main result establishes exhaustive and explicit complexity classifications for counting subgraphs and induced subgraphs. We show that the (not necessarily induced) copies of $H$ in $G$ can be counted in time $f(k,d)\cdot n^{\max(\mathsf{imn}(H),1)}\cdot \log n$, where $f$ is some computable function and $\mathsf{imn}(H)$ is the size of the largest induced matching of $H$. Whenever the class of allowed patterns has unbounded induced matching number, this algorithm is essentially optimal: Unless the Exponential Time Hypothesis (ETH) fails, there is no algorithm running in time $f(k,d)\cdot n^{o(\mathsf{imn}(H)/\log \mathsf{imn}(H))}$ for any function $f$. In case of counting induced subgraphs, we obtain a similar classification along the independence number $\alpha$: we can count the induced copies of $H$ in $G$ in time $f(k,d)\cdot n^{\alpha(H)}\cdot \log n$, and if the class of allowed patterns has unbounded independence number, an algorithm running in time $f(k,d)\cdot n^{o(\alpha(H)/\log \alpha(H))}$ is impossible, unless ETH fails. In the language of parameterized complexity, our results yield dichotomies in fixed-parameter tractable and $\#\mathsf{W}[1]$-hard cases if we parameterize by the size of the pattern and the degeneracy of the host graph. Our results imply that several patterns cannot be counted in time $f(k,d)\cdot n^{o(k/\log k)}$, including $k$-matchings, $k$-independent sets, (induced) $k$-paths, (induced) $k$-cycles, and induced $(k,k)$-bicliques, unless ETH fails. Those lower bounds for exact counting are complemented with new algorithms for approximate counting of subgraphs and induced subgraphs in degenerate graphs.