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Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs

Abstract

We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, {\sc kk-Leaf Out-Branching}, which is to find an oriented spanning tree with at least kk leaves, we obtain an algorithm solving the problem in time 2O(klogk)n+nO(1)2^{O(\sqrt{k} \log k)} n+ n^{O(1)} on directed graphs whose underlying undirected graph excludes some fixed graph HH as a minor. For the special case when the input directed graph is planar, the running time can be improved to 2O(k)n+nO(1)2^{O(\sqrt{k})}n + n^{O(1)}. The second example is a generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc kk-Internal Out-Branching}, which is to find an oriented spanning tree with at least kk internal vertices. We obtain an algorithm solving the problem in time 2O(klogk)+nO(1)2^{O(\sqrt{k} \log k)} + n^{O(1)} on directed graphs whose underlying undirected graph excludes some fixed apex graph HH as a minor. Finally, we observe that for any ϵ>0\epsilon>0, the {\sc kk-Directed Path} problem is solvable in time O((1+ϵ)knf(ϵ))O((1+\epsilon)^k n^{f(\epsilon)}), where ff is some function of \ve. Our methods are based on non-trivial combinations of obstruction theorems for undirected graphs, kernelization, problem specific combinatorial structures and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs

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