Cut and Count and Representative Sets on Branch Decompositions

Authors Willem J. A. Pino, Hans L. Bodlaender, Johan M. M. van Rooij



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Willem J. A. Pino
Hans L. Bodlaender
Johan M. M. van Rooij

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Willem J. A. Pino, Hans L. Bodlaender, and Johan M. M. van Rooij. Cut and Count and Representative Sets on Branch Decompositions. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 27:1-27:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.IPEC.2016.27

Abstract

Recently, new techniques have been introduced to speed up dynamic programming algorithms on tree decompositions for connectivity problems: the 'Cut and Count' method and a method called the rank-based approach, based on representative sets and Gaussian elimination. These methods respectively give randomised and deterministic algorithms that are single exponential in the treewidth, and polynomial, respectively linear in the number of vertices. In this paper, we adapt these methods to branch decompositions yielding algorithms, both randomised and deterministic, that are in many cases faster than when tree decompositions would be used.

In particular, we obtain the currently fastest randomised algorithms for several problems on planar graphs. When the involved weights are O(n^{O(1)}), we obtain faster randomised algorithms on planar graphs for Steiner Tree, Connected Dominating Set, Feedback Vertex Set and TSP, and a faster deterministic algorithm for TSP. When considering planar graphs with arbitrary real weights, we obtain faster deterministic algorithms for all four mentioned problems.

Subject Classification

Keywords
  • Graph algorithms
  • Branchwidth
  • Treewidth
  • Dynamic Programming
  • Planar Graphs

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References

  1. Hans L. Bodlaender. Treewidth: Structure and algorithms. In Proceedings of the 14th International Colloquium on Structural Information and Communication Complexity, SIROCCO 2007, volume 4474 of Lecture Notes in Computer Science, pages 11-25. Springer Verlag, 2007. Google Scholar
  2. Hans L. Bodlaender, Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Information and Computation, 243:86-111, 2015. Google Scholar
  3. Hans L. Bodlaender, Erik Jan van Leeuwen, Johan M. M. van Rooij, and Martin Vatshelle. Faster algorithms on branch and clique decompositions. In Proceedings of the 35th International Symposium on Mathematical Foundations of Computer Science, MFCS 2010, volume 6281 of Lecture Notes in Computer Science, pages 174-185. Springer Verlag, 2010. Google Scholar
  4. Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Fast Hamiltonicity checking via bases of perfect matchings. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 301-310. ACM, 2013. Google Scholar
  5. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 150-159, 2011. Google Scholar
  6. Frederic Dorn. Dynamic programming and fast matrix multiplication. In Proceedings of the 14th Annual European Symposium on Algorithms, ESA 2006, volume 4168 of Lecture Notes in Computer Science, pages 280-291. Springer Verlag, 2006. Google Scholar
  7. Frederic Dorn. Designing Subexponential Algorithms: Problems, Techniques &Structures. PhD thesis, Institutt for informatikk, Universitetet i Bergen, 2007. Google Scholar
  8. Frederic Dorn, Eelko Penninkx, Hans L. Bodlaender, and Fedor V. Fomin. Efficient exact algorithms on planar graphs: exploiting sphere cut decompositions. Algorithmica, 58:790-810, 2010. Google Scholar
  9. Stefan Fafianie, Hans L. Bodlaender, and Jesper Nederlof. Speeding up dynamic programming with representative sets: an experimental evaluation of algorithms for steiner tree on tree decompositions. Algorithmica, 71(3):636-660, 2015. Google Scholar
  10. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Representative sets of product families. In Algorithms - ESA 2014 - 22th Annual European Symposium, Wroclaw, Poland, September 8-10, 2014. Proceedings, volume 8737 of Lecture Notes in Computer Science, pages 443-454. Springer, 2014. Google Scholar
  11. Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Efficient computation of representative sets with applications in parameterized and exact algorithms. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, pages 142-151, 2014. Google Scholar
  12. Fedor V. Fomin and Dimitrios M. Thilikos. New upper bounds on the decomposability of planar graphs. Journal of Graph Theory, 51:53-81, 2006. Google Scholar
  13. François Le Gall. Faster algorithms for rectangular matrix multiplication. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 514-523. IEEE Computer Society, 2012. Google Scholar
  14. François Le Gall. Powers of tensors and fast matrix multiplication. In International Symposium on Symbolic and Algebraic Computation, ISSAC, pages 296-303, 2014. Google Scholar
  15. Qian-Ping Gu and Hisao Tamaki. Optimal branch-decomposition of planar graphs in O(n³) time. ACM Trans. Algorithms, 4(3), 2008. Google Scholar
  16. Paul D. Seymour and Robin Thomas. Call routing and the ratcatcher. Combinatorica, 14(2):217-241, 1994. Google Scholar
  17. Jan Arne Telle and Andrzej Proskurowski. Algorithms for vertex partitioning problems on partial k-trees. SIAM J. Discr. Math., 10:529-550, 1997. Google Scholar
  18. Johan M. M. van Rooij, Hans L. Bodlaender, and Peter Rossmanith. Dynamic programming on tree decompositions using generalised fast subset convolution. In Proceedings of the 17th Annual European Symposium on Algorithms, ESA 2009, volume 5757 of Lecture Notes in Computer Science, pages 566-577. Springer Verlag, 2009. Google Scholar
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