Faster Steiner Tree Computation in Polynomial-Space

Given an n-node graph and a subset of k terminal nodes, the NP -hard Steiner tree problem is to compute a minimum-size tree which spans the terminals. All the known algorithms for this problem which improve on trivial O(1.62)-time enumeration are based on dynamic programming, and require exponential space. Motivated by the fact that exponential-space algorithms are typically impractical, in this paper we address the problem of designing faster polynomial-space algorithms. Our first contribution is a simple polynomialspace O(6n )-time algorithm, based on a variant of the classical tree-separator theorem. This improves on trivial O(n) enumeration for, roughly, k ≤ n/4. Combining the algorithm above (for small k), with an improved branching strategy (for large k), we obtain an O(1.60)-time polynomial-space algorithm. The refined branching is based on a charging mechanism which shows that, for large values of k, convenient local configurations of terminals and non-terminals must exist. The analysis of the algorithm relies on the Measure & Conquer approach: the non-standard measure used here is a linear combination of the number of nodes and number of non-terminals. As a byproduct of our work, we also improve the (exponential-space) time complexity of the problem from O(1.42) to O(1.36).