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

Abstract

We present an algorithm to count the number of occurrences of a pattern graph H on h vertices as an induced subgraph in a host graph G. If G belongs to a bounded expansion class, the algorithm runs in linear time, if G belongs to a nowhere dense class it runs in almost-linear time. Our design choices are motivated by the need for an approach that can be engineered into a practical implementation for sparse host graphs. 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 $O(\Vert \vec {C}\Vert \cdot h {\text {wcol}}_{h} (\mathbb {G})^{h-1} |G|)$, 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 $O(\Vert \vec {C}\Vert \cdot h {\text {col}}_{h} (\mathbb {G})^{h-1} |G|)$, 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.