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

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 C⃗(H) which encodes an order-aware, inclusion-exclusion counting method for H . Given such a counting dag and a suitable linear ordering 𝔾 of G as input, our algorithm can count the number of times H appears as an induced subgraph in G in time O(‖C⃗‖· h wcol_h (𝔾)^h-1 |G|) , where wcol_h (𝔾) denotes the maximum size of the weakly h -reachable sets in 𝔾 . This implies, combined with previous results, an algorithm with running time O((3h^2 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(‖C⃗‖· h col_h (𝔾)^h-1 |G|) , resulting in an overall complexity of O(h (3 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.