Exact and Parameterized Algorithms for Max Internal Spanning Tree

摘要

We consider the NP-hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form O~*(c") with c ≤ 2. For graphs with bounded degree, c ＜ 2. The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of O(1.8612") when analyzed with respect to the number of vertices. We also show that its running time is 2.1364~kn~(O(1)) when the goal is to find a spanning tree with at least k internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analysis for parameterized algorithms.