A color-avoiding approach to subgraph counting in bounded expansion classes
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We present an algorithm to count the number of occurrences of a pattern graph H as an induced subgraph in a host graph G. If G belongs to a bounded expansion class, the algorithm runs in 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(4^h^2h (wcol_h(G)+1)^h^3 |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(4^h^2h 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 [11], our algorithm provides a promising alternative to algorithms using the traditional p-treedepth colouring framework [13]. We describe preliminary experimental results from an initial open source implementation which highlight its potential.
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