A Color-Avoiding Approach to Subgraph Counting in Bounded Expansion Classes.

Specifically, we introduce a decomposition of the pattern $H$ called a counting dag $\vec C(H)$ which encodes an order-aware, inclusion-exclusion counting method for $H$. Given such a counting dag and a suitable linear ordering $\mathbb{G}$ of $G$ as input, our algorithm can count the number of times $H$ appears as an induced subgraph in $G$ in time $O(|\vec C| \cdot h \text{wcol}_{h}(\mathbb{G})^{h-1} |G|)$, where ${\text{wcol}}h(\mathbb{G})$ denotes the maximum size of the weakly $h$-reachable sets in $\mathbb{G}$. This implies, combined with previous results, an algorithm with running time $O((3h^2 \text{wcol}{h}(G))^{h^2} |G|)$ which only takes $H$ and $G$ as input.

We note that with a small modification, our algorithm can instead use strongly $h$-reachable sets with running time $O(|\vec C| \cdot h \text{col}_{h}(\mathbb{G})^{h-1} |G|)$, resulting in an overall complexity of $O(h (3\text{col}_h(G))^{h^2} |G|)$ when only given $H$ and $G$.

Because orderings with small weakly/strongly reachable sets can be computed relatively efficiently in practice (Nadara et al.: in J Exp Algorithmics 103:14:1–14:16, 2018), our algorithm provides a promising alternative to algorithms using the traditional $p$-treedepth coloring framework (O’Brien and Sullivan in: Experimental evaluation of counting subgraph isomorphisms in classes of bounded expansion, CoRR, arXiv:1712.06690, 2017). We describe preliminary experimental results from an initial open source implementation which highlight its potential.