Kernelization for Maximum Leaf Spanning Tree with Positive Vertex Weights

In this paper we consider a natural generalization of the well-known Max Leaf Spanning Tree problem. In the generalized Weighted Max Leaf problem we get as input an undirected connected graph G, a rational number k not smaller than 1 and a weight function w : V 7→ Q≥1 on the vertices, and are asked whether a spanning tree T forG exists such that the combined weight of the leaves of T is at least k. We show that it is possible to transform an instance 〈G,w, k〉 of Weighted Max Leaf in polynomial time into an equivalent instance 〈G′, w′, k′〉 such that |V (G′)| ≤ 5.5k and k′ ≤ k. In the context of parameterized complexity this means that Weighted Max Leaf admits a kernel with 5.5k vertices. The analysis of the kernel size is based on a new extremal result which shows that every graph G = (V,E) that excludes some simple substructures always contains a spanning tree with at least |V |/5.5 leaves. We also prove that Weighted Max Leaf does not admit a polynomial-time factor O(n 1 2 −ε) or O(opt 1 3 −ε) approximation algorithm for any ε > 0 unless P = NP. Submitted: September 2011 Reviewed: May 2012 Revised: July 2012 Accepted: October 2012 Final: October 2012 Published: October 2012 Article type: Regular Paper Communicated by: G. Woeginger This work was supported by the Netherlands Organization for Scientific Research (NWO), project “KERNELS: Combinatorial Analysis of Data Reduction”. A preliminary version appeared at the 7th International Conference on Algorithms and Complexity (CIAC 2010). E-mail address: [email protected] (Bart M. P. Jansen) 812 Bart M. P. Jansen Kernelization for Weighted Max Leaf