Parameterized Complexity of Directed Steiner Tree and Domination Problems on Sparse Graphs
Ondřej Suchý (Dr., TU Berlin)
We study the parameterized complexity of the directed variant of the classical Steiner Tree problem on various classes of sparse graphs. While the parameterized complexity of Steiner Tree on undirected graphs parameterized by the number of terminals is well understood, not much is known about the parameterization by the number of non-terminals in the solution tree. All that is known for this parameterization is that both the directed and the undirected versions are W-hard on general graphs, and hence unlikely to be fixed parameter tractable (FPT). The undirected Steiner Tree problem becomes FPT when restricted to sparse classes of graphs such as planar graphs, but the techniques used to show this result break down on directed planar graphs.
In this talk we precisely chart the tractability border for Directed Steiner Tree (DST) on sparse graphs parameterized by the number of non-terminals in the solution tree. Specifically, we show that the problem is fixed parameter tractable on graphs excluding a topological minor, but becomes W-hard on graphs of degeneracy 2. On the other hand we show that if the subgraph induced by the terminals is required to be acyclic then the problem becomes FPT on graphs of bounded degeneracy.
Our algorithms for DST are based on a novel branching rule. To
demonstrate the versatility of the new branching we use it to give
improved parameterized algorithms for Dominating Set on graphs of
bounded degeneracy and graphs excluding a topological minor. We
further show that our algorithm achieves the best possible asymptotic
running time dependence on the solution size and degeneracy of the
input graph, under standard complexity theoretic assumptions.
This is a joint work with Mark Jones, Daniel Lokshtanov, M. S. Ramanujan and Saket Saurabh.
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