The Complexity of Pattern Counting in Directed Graphs, Parameterised by the Outdegree
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We study the fixed-parameter tractability of the following fundamental problem: given two directed graphs H⃗ and G⃗, count the number of copies of H⃗ in G⃗. The standard setting, where the tractability is well understood, uses only |H⃗| as a parameter. In this paper we take a step forward, and adopt as a parameter |H⃗|+d(G⃗), where d(G⃗) is the maximum outdegree of |G⃗|. Under this parameterization, we completely characterize the fixed-parameter tractability of the problem in both its non-induced and induced versions through two novel structural parameters, the fractional cover number ρ^* and the source number α_s. On the one hand we give algorithms with running time f(|H⃗|,d(G⃗)) · |G⃗|^ρ^*(H⃗)+O(1) and f(|H⃗|,d(G⃗)) · |G⃗|^α_s(H⃗)+O(1) for counting respectively the copies and induced copies of H⃗ in G⃗; on the other hand we show that, unless the Exponential Time Hypothesis fails, for any class C⃗ of directed graphs the (induced) counting problem is fixed-parameter tractable if and only if ρ^*(C⃗) (α_s(C⃗)) is bounded. These results explain how the orientation of the pattern can make counting easy or hard, and prove that a classic algorithm by Chiba and Nishizeki and its extensions (Chiba, Nishizeki SICOMP 85; Bressan Algorithmica 21) are optimal unless ETH fails.
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