an:06464056
Zbl 1317.05072
Kammer, Frank
A linear-time kernelization for the rooted \(k\)-leaf outbranching problem
EN
Discrete Appl. Math. 193, 126-138 (2015).
00346652
2015
j
05C20 05C05 05C69 68Q25
parameterized complexity; linear-time data reduction; NP-hard graph problems; dominator trees
Summary: In the rooted \(k\)-leaf outbranching problem, a digraph \(G = (V, E)\), a vertex \(r\) of \(G\), and an integer \(k\) are given, and the goal is to find an \(r\)-rooted spanning outtree of \(G\) with \(\geq k\) leaves (a subtree of \(G\) with vertex set \(V\), all edges directed away from \(r\), and \(\geq k\) leaves). We present a linear-time algorithm that computes a problem kernel with \(O(k^6)\) vertices and \(O(k^7)\) edges for the rooted \(k\)-leaf outbranching problem. By combining the new result with a result of \textit{J. Daligault} and \textit{S. Thomass??} [Lect. Notes Comput. Sci. 5917, 86--97 (2009; Zbl 1273.68162)], a kernel with a quadratic number of vertices and edges can be found on \(n\)-vertex \(m\)-edge digraphs in time \(O(n + m + k^{14})\).
Zbl 1273.68162