Faster Subgraph Counting in Sparse Graphs

Abstract

A fundamental graph problem asks to compute the number of induced copies of a $k$-node pattern graph $H$ in an $n$-node graph $G$. The fastest algorithm to date is still the 35-years-old algorithm by Ne\v{s}et\v{r}il and Poljak [31], with running time $f(k) \cdot O(n^{\omega\lfloor\frac{k}{3}\rfloor + 2})$ where $\omega \le 2.373$ is the matrix multiplication exponent. In this work we show that, if one takes into account the degeneracy $d$ of $G$, then the picture becomes substantially richer and leads to faster algorithms when $G$ is sufficiently sparse. More precisely, after introducing a novel notion of graph width, the \emph{DAG-treewidth}, we prove what follows. If $H$ has DAG-treewidth $\tau(H)$ and $G$ has degeneracy $d$, then the induced copies of $H$ in $G$ can be counted in time $f(d,k) \cdot \tilde{O}(n^{\tau(H)})$; and, under the Exponential Time Hypothesis, no algorithm can solve the problem in time $f(d,k) \cdot n^{o(\tau(H)/\ln \tau(H))}$ for all $H$. This result characterises the complexity of counting subgraphs in a $d$-degenerate graph. Developing bounds on $\tau(H)$, then, we obtain natural generalisations of classic results and faster algorithms for sparse graphs. For example, when $d=O(\operatorname{poly}\log(n))$ we can count the induced copies of any $H$ in time $f(k) \cdot\tilde{O}(n^{\lfloor \frac{k}{4} \rfloor + 2})$, beating the Ne\v{s}et\v{r}il-Poljak algorithm by essentially a cubic factor in $n$.