Extending the Kernel for Planar Steiner Tree to the Number of Steiner Vertices

Abstract

In the Steiner Tree problem one is given an undirected graph, a subset T of its vertices, and an integer k and the question is whether there is a connected subgraph of the given graph containing all the vertices of T and at most k other vertices. The vertices in the subset T are called terminals and the other vertices are called Steiner vertices. Recently, Pilipczuk, Pilipczuk, Sankowski, and van Leeuwen [FOCS 2014] gave a polynomial kernel for Steiner Tree in planar graphs, when parameterized by |T | + k, the total number of vertices in the constructed subgraph.In this paper we present several polynomial time applicable reduction rules for Planar Steiner Tree. In an instance reduced with respect to the presented reduction rules, the number of terminals |T | is at most quadratic in the number of other vertices k in the subgraph. Hence, using and improving the result of Pilipczuk et al., we give a polynomial kernel for Steiner Tree in planar graphs for the parameterization by the number k of Steiner vertices in the solution.

Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

“…The idea is that computing the kernel in this case is an efficient preprocessing procedure for the problem, such that exhaustive search algorithms can be used on the kernel. Suchý [33] proved that Unweighted Steiner Tree parameterized by p admits a polynomial kernel if the input graph is planar.…”

“…The vertex resulting from the contraction is declared a terminal and the process repeats for the new graph. Previous results [24,33] have also built on this straightforward procedure in order to obtain FPT algorithms and polynomial kernels for special cases of Unweighted Directed Steiner Tree and Unweighted Steiner Tree. In particular, in the unweighted undirected setting it is a well-known fact (cf.…”

“…In particular, in the unweighted undirected setting it is a well-known fact (cf. [33]) that contracting an adjacent pair of terminals is always a safe option, as there always exists an optimum Steiner tree containing this edge. However this immediately breaks down if the input graph is edge-weighted, as an edge between terminals might be very costly and should therefore not be contained in any (approximate) solution.…”

“…This parameter p has been mainly studied in the context of unweighted graphs before. The problem remains W [2]-hard in this special case and therefore the focus has been on designing parameterized algorithms for restricted graph classes, such as planar or d-degenerate graphs [24,33].…”

On Subexponential Parameterized Algorithms for Steiner Tree and Directed Subset TSP on Planar Graphs

“…In fact, this question has been posed as a natural open problem in various places [6,11,19,34,38,39]. As partial progress toward this goal, in the unweighted case, subexponential algorithms parameterized by the number of edges of the solution and number of nonterminal vertices were found [38,39,41]. However, the number of edges can be of course much larger than the number of terminals, hence an algorithm that is subexponential in the number of edges is not necessarily subexponential in the number of terminals.…”