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DOI: 10.4230/LIPIcs.IPEC.2017.16

K-Best Solutions of MSO Problems on Tree-Decomposable Graphs

Authors: David Eppstein and Denis Kurz

Published in: LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)

Abstract

We show that, for any graph optimization problem in which the feasible solutions can be expressed by a formula in monadic second-order logic describing sets of vertices or edges and in which the goal is to minimize the sum of the weights in the selected sets, we can find the k best solution values for n-vertex graphs of bounded treewidth in time O(n + k log n). In particular, this applies to finding the k shortest simple paths between given vertices in directed graphs of bounded treewidth, giving an exponential speedup in the per-path cost over previous algorithms.

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David Eppstein and Denis Kurz. K-Best Solutions of MSO Problems on Tree-Decomposable Graphs. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 16:1-16:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{eppstein_et_al:LIPIcs.IPEC.2017.16,
  author =	{Eppstein, David and Kurz, Denis},
  title =	{{K-Best Solutions of MSO Problems on Tree-Decomposable Graphs}},
  booktitle =	{12th International Symposium on Parameterized and Exact Computation (IPEC 2017)},
  pages =	{16:1--16:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-051-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{89},
  editor =	{Lokshtanov, Daniel and Nishimura, Naomi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.16},
  URN =		{urn:nbn:de:0030-drops-85494},
  doi =		{10.4230/LIPIcs.IPEC.2017.16},
  annote =	{Keywords: graph algorithm, k-best, monadic second-order logic, treewidth}
}

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DOI: 10.4230/LIPIcs.ISAAC.2016.49

A Sidetrack-Based Algorithm for Finding the k Shortest Simple Paths in a Directed Graph

Authors: Denis Kurz and Petra Mutzel

Published in: LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

Abstract

We present an algorithm for the k shortest simple path problem on weighted directed graphs (kSSP) that is based on Eppstein’s algorithm for a similar problem in which paths are allowed to contain cycles. In contrast to most other algorithms for kSSP, ours is not based on Yen's algorithm [Networks, 1971] and does not solve replacement path problems. Its worst-case running time is on par with state-of-the-art algorithms for kSSP. Using our algorithm, one may find O(m) simple paths with a single shortest path tree computation and O(n+m) additional time per path in well-behaved cases, where n is the number of nodes and m is the number of edges. Our computational results show that on random graphs and large road networks, these well-behaved cases are quite common and our algorithm is faster than existing algorithms by an order of magnitude.

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Denis Kurz and Petra Mutzel. A Sidetrack-Based Algorithm for Finding the k Shortest Simple Paths in a Directed Graph. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 49:1-49:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kurz_et_al:LIPIcs.ISAAC.2016.49,
  author =	{Kurz, Denis and Mutzel, Petra},
  title =	{{A Sidetrack-Based Algorithm for Finding the k Shortest Simple Paths in a Directed Graph}},
  booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
  pages =	{49:1--49:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-026-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{64},
  editor =	{Hong, Seok-Hee},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.49},
  URN =		{urn:nbn:de:0030-drops-68199},
  doi =		{10.4230/LIPIcs.ISAAC.2016.49},
  annote =	{Keywords: directed graph, k-best, shortest path, simple path, weighted graph}
}

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Documents authored by Kurz, Denis

K-Best Solutions of MSO Problems on Tree-Decomposable Graphs

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A Sidetrack-Based Algorithm for Finding the k Shortest Simple Paths in a Directed Graph

Abstract

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