Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs

In 2000 Alber et al. [SWAT 2000 ] obtained the first parameterized subexponential algorithm on undirected planar graphs by showing that k-DOMINATING SET is solvable in time 2O( √ k)nO(1), where n is the input size. This result triggered an extensive study of parameterized problems on planar and more general classes of sparse graphs and culminated in the creation of Bidimensionality Theory by Demaine et al. [J. ACM 2005 ]. The theory utilizes deep theorems from Graph Minor Theory of Robertson and Seymour, and provides a simple criteria for checking whether a parameterized problem is solvable in subexponential time on sparse graphs. While bidimensionality theory is an algorithmic framework on undirected graphs, it remains unclear how to apply it to problems on directed graphs. The main reason is that Graph Minor Theory for directed graphs is still in a nascent stage and there are no suitable obstruction theorems so far. Even the analogue of treewidth for directed graphs is not unique and several alternative definitions have been proposed. In this paper we make the first step beyond bidimensionality by obtaining subexponential time algorithms for problems on directed graphs. 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, k-LEAF OUT-BRANCHING, which is to find an oriented spanning tree with at least k leaves, we obtain an algorithm solving the problem in time 2O( √ k log n + nO(1) on directed graphs whose underlying undirected graph excludes some fixed graph H as a minor. For the special case when the input directed graph is planar, the running time can be improved to 2O( √ n + nO(1). The second example is a generalization of the DIRECTED HAMILTONIAN PATH problem, namely k-INTERNAL OUT-BRANCHING, which is to find an oriented spanning tree with at least k internal vertices. We obtain an algorithm solving the problem in time 2O( √ k log k)+nO(1) on directed graphs whose underlying undirected graph excludes some fixed apex graph H as a minor. Finally, we observe that for any ε > 0, the k-DIRECTED PATH problem is solvable in time O((1 + ε)n), where f is some function of ε. 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.