A Slice Theoretic Approach for Embedding Problems on Digraphs

Abstract

We say that a digraph H can be covered by k paths if there exist k directed paths \(\mathfrak {p}_1,\mathfrak {p}_2,\ldots ,\mathfrak {p}_k\) such that \(H=\cup _{i=1}^k \mathfrak {p}_i\). In this work we devise parameterized algorithms for embedding problems on digraphs in the setting in which the host digraph G has directed pathwidth w and the pattern digraph H can be covered by k paths. More precisely, we show that the subgraph isomorphism, subgraph homeomorphism, and two other related embedding problems can each be solved in time \(2^{O(k\cdot w \log k\cdot w)} \cdot |H|^{O(k\cdot w)}\cdot |G|^{O(k\cdot w)}\). We note in particular that for constant values of w and k, our algorithm runs in polynomial time with respect to the size of the pattern digraph H. Therefore for the classes of digraphs considered in this work our results yield an exponential speedup with respect to the best general algorithm for the subgraph isomorphism problem which runs in time \(O^*(2^{|H|}\cdot |G|^{ tw (H)})\) (where \( tw (H)\) is the undirected treewidth of H), and an exponential speedup with respect to the best general algorithm for the subgraph homeomorphism problem which runs in time \(|G|^{O(|H|)}\).