Computing Optimal Steiner Trees in Polynomial Space

• Published in: Algorithmica Vol. 65; no. 3; pp. 584 - 604
• Main Authors: Fomin, Fedor V., Grandoni, Fabrizio, Kratsch, Dieter, Lokshtanov, Daniel, Saurabh, Saket
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.03.2013; Springer
• Subjects: Algorithm Analysis and Problem Complexity; Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Computer Science; Computer science; control theory; systems; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Exact sciences and technology; Information retrieval. Graph; Mathematics of Computing; Theoretical computing; Theory of Computation; Steiner tree; Space complexity; Exact algorithms; Branching; Edge(graph); Tree(graph); Enumeration; Automaton; Logarithmic function; Algorithmics; Steiner tree problem; Computational complexity; Terminal; Weighted graph; Polynomial time; Programming language; NP hard problem; Exponential time; Dynamic programming; Exact algorithm; Algorithm analysis; Separator
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-012-9612-z

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 2 k /3 terminals each. Exploiting separators of logarithmic size which evenly partition the terminals, we are able to reduce the running time to . 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 (International colloquium on automata, languages and programming (ICALP), pp. 713–725, 2009 ), 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.