Computing Optimal Steiner Trees in Polynomial Space

Abstract

Given an n-node edge-weighted graph and a subset of k terminal nodes, the NP-hard (weighted) Steiner tree problem is to compute a minimum-weight tree which spans the terminals. All the known algorithms for this problem which improve on trivial O(1.62 n )-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 O((27/4) k n O(log k) )-time polynomial-space algorithm for the problem. This algorithm is based on a variant of the classical tree-separator theorem: every Steiner tree has a node whose removal partitions the tree in two forests, containing at most 2k/3 terminals each. Exploiting separators of logarithmic size which evenly partition the terminals, we are able to reduce the running time to O(4 k n O(log 2 k) ). This improves on trivial enumeration for roughly k < n/3, which covers most of the cases of practical interest. Combining the latter algorithm (for small k) with trivial enumeration (for large k) we obtain a O(1.59 n )-time polynomial-space algorithm for the weighted Steiner tree problem.As a second contribution of this paper, we present a O(1.55 n )-time polynomial-space algorithm for the cardinality version of the problem, where all edge weights are one. This result is based on a improved branching strategy. 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 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. Using a recent result in [Nederlof'09], the running time can be reduced to O(1.36 n ). The previous best algorithm for the cardinality case runs in O(1.42 n ) time and exponential space.

Parameterized Single-Exponential Time Polynomial Space Algorithm for Steiner Tree

“…For weighted Steiner Tree, the only known polynomial space FPT algorithm has a 2 O(k log k) running time dependence on the parameter k. This algorithm follows from combining a (27/4) k · n O(log k) · log W time, polynomial space algorithm by Fomin et al [9] with the Dreyfus-Wagner algorithm. Indeed, one runs the algorithm of Fomin et al [9] if n ≤ 2 k , and the Dreyfus-Wagner algorithm if n > 2 k .…”

“…Indeed, one runs the algorithm of Fomin et al [9] if n ≤ 2 k , and the Dreyfus-Wagner algorithm if n > 2 k . If n ≤ 2 k , the running time of the algorithm of Fomin et al is bounded from above by 2 O(k log k) .…”

“….e a single exponential time polynomial space FPT algorithm, was an open problem asked explicitly in [9,12].…”

“…The starting point of our present algorithm is the (27/4) k · n O(log k) · log W -time, polynomial-space algorithm by Fomin et al [9]. This algorithm crucially exploits the possibility for balanced separation (cf.…”

Inexact graph matching using a hierarchy of matching processes

“…The high computational complexity of current optimal methods poses a significant problem, which we attempt to address in this paper. Indeed, there has been some recent interest in reducing the computational complexity of inherently NP-hard problems, particularly in the field of graph theory [16][17][18][19][20].…”