Fast Polynomial-Space Algorithms Using Inclusion-Exclusion
Jesper Nederlof · Algorithmica · 2012
Given a graph with n vertices, k terminals and positive integer weights not larger than c, we compute a minimum Steiner Tree in $\mathcal{O}^{\star}(2^{k}c)$ time and $\mathcal{O}^{\star}(c)$ space, where the $\mathcal{O}^{\star}$ notation omits terms bounded by a polynomial in the input-size. We obtain the result by defining a generalization of walks, called branching walks, and combining it with the Inclusion-Exclusion technique. Using this combination we also give $\mathcal{O}^{\star}(2^{n})$ -time polynomial space algorithms for Degree Constrained Spanning Tree, Maximum Internal Spanning Tree and #Spanning Forest with a given number of components. Furthermore, using related techniques, we also present new polynomial space algorithms for computing the Cover Polynomial of a graph, Convex Tree Coloring and counting the number of perfect matchings of a graph.