Algorithmica

, Volume 65, Issue 3, pp 584–604 | Cite as

Computing Optimal Steiner Trees in Polynomial Space

  • Fedor V. Fomin
  • Fabrizio Grandoni
  • Dieter Kratsch
  • Daniel Lokshtanov
  • Saket Saurabh
Article

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(logk))-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 (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.

Keywords

Steiner tree Exact algorithms Space complexity 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Fedor V. Fomin
    • 1
  • Fabrizio Grandoni
    • 2
  • Dieter Kratsch
    • 3
  • Daniel Lokshtanov
    • 1
  • Saket Saurabh
    • 4
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.IDSIAUniversity of Italian SwitzerlandMannoSwitzerland
  3. 3.LITAUniversité Paul Verlaine – MetzMetz Cedex 01France
  4. 4.The Institute of Mathematical SciencesTaramaniChennaiIndia

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