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Faster Exact Algorithms for Some Terminal Set Problems

  • Conference paper
Parameterized and Exact Computation (IPEC 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8246))

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Abstract

Many problems on graphs can be expressed in the following language: given a graph G = (V,E) and a terminal set T ⊆ V, find a minimum size set S ⊆ V which intersects all “structures” (such as cycles or paths) passing through the vertices in T. We call this class of problems as terminal set problems. In this paper we introduce a general method to obtain faster exact exponential time algorithms for many terminal set problems. More precisely, we show that

  • Node Multiway Cut can be solved in time O(1.4766n).

  • Directed Unrestricted Node Multiway Cut can be solved in time O(1.6181n).

  • There exists a deterministic algorithm for Subset Feedback Vertex Set running in time O(1.8980n) and a randomized algorithm with expected running time O(1.8826n). Furthermore, Subset Feedback Vertex on chordal graphs can be solved in time O(1.6181n).

  • Directed Subset Feedback Vertex Set can be solved in time O(1.9993n).

A key feature of our method is that, it uses the existing best polynomial time, fixed parameter tractable and exact exponential time algorithms for the non-terminal version of the same problem (i.e. when T = V), as subroutines. Therefore faster algorithms for these special cases will imply further improvements in the running times of our algorithms. Our algorithms for Node Multiway Cut, and Subset Feedback Vertex Set on chordal graphs improve the current best algorithms for these problems and answers an open question posed in [15]. Furthermore, our algorithms for Directed Unrestricted Node Multiway Cut and Directed Subset Feedback Vertex Set are the first exact algorithms improving upon the brute force O *(2n)-algorithms.

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References

  1. Belmonte, R., Golovach, P.A., Heggernes, P., van ’t Hof, P., Kamiński, M., Paulusma, D.: Finding contractions and induced minors in chordal graphs via disjoint paths. In: Asano, T., Nakano, S.-i., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 110–119. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  2. Cao, Y., Chen, J., Liu, Y.: On feedback vertex set new measure and new structures. In: Kaplan, H. (ed.) SWAT 2010. LNCS, vol. 6139, pp. 93–104. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  3. Chen, J., Fomin, F.V., Liu, Y., Lu, S., Villanger, Y.: Improved algorithms for feedback vertex set problems. J. Comput. Syst. Sci. 74(7), 1188–1198 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theoretical Computer Science 411(40), 3736–3756 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, J., Liu, Y., Lu, S.: An Improved Parameterized Algorithm for the Minimum Node Multiway Cut Problem. Algorithmica 55(1), 1–13 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Corneil, D., Fonlupt, J.: The complexity of generalized clique covering. Discrete Applied Mathematics 22(2), 109–118 (1988–1989)

    Google Scholar 

  7. Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J., Wojtaszczyk, J.: Solving connectivity problems parameterized by treewidth in single exponential time. In: 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 150–159. IEEE (2011)

    Google Scholar 

  8. Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: On multiway cut parameterized above lower bounds. In: Marx, D., Rossmanith, P. (eds.) IPEC 2011. LNCS, vol. 7112, pp. 1–12. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  9. Fomin, F.V., Gaspers, S., Pyatkin, A.V., Razgon, I.: On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52(2), 293–307 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fomin, F.V., Heggernes, P., Kratsch, D., Papadopoulos, C., Villanger, Y.: Enumerating minimal subset feedback vertex sets. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 399–410. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  11. Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms, 1st edn. Springer-Verlag New York, Inc., New York (2010)

    Book  MATH  Google Scholar 

  12. Fomin, F.V., Villanger, Y.: Finding induced subgraphs via minimal triangulations. In: 27th International Symposium on Theoretical Aspects of Computer Science (STACS), vol. 5, pp. 383–394. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2010)

    Google Scholar 

  13. Fomin, F.V., Villanger, Y.: Finding induced subgraphs via minimal triangulations. In: STACS, vol. 5, pp. 383–394 (2010)

    Google Scholar 

  14. Garg, N., Vazirani, V., Yannakakis, M.: Multiway Cuts in Directed and Node Weighted Graphs. In: Shamir, E., Abiteboul, S. (eds.) ICALP 1994. LNCS, vol. 820, pp. 487–498. Springer, Heidelberg (1994)

    Chapter  Google Scholar 

  15. Golovach, P.A., Heggernes, P., Kratsch, D., Saei, R.: An exact algorithm for subset feedback vertex set on chordal graphs. In: Thilikos, D.M., Woeginger, G.J. (eds.) IPEC 2012. LNCS, vol. 7535, pp. 85–96. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  16. Gupta, S., Raman, V., Saurabh, S.: Maximum r-regular induced subgraph problem: Fast exponential algorithms and combinatorial bounds. SIAM J. Discrete Math. 26(4), 1758–1780 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kratsch, D., Müller, H., Todinca, I.: Feedback vertex set on at-free graphs. Discrete Applied Mathematics 156(10), 1936–1947 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mader, W.: Über die Maximalzahl kreuzungsfreier H-Wege. Arch. Math (Basel) 31(4), 387–402 (1978/1979), http://dx.doi.org/10.1007/BF01226465

    Google Scholar 

  19. Marx, D.: Parameterized Graph Separation Problems. Theor. Comput. Sci. 351(3), 394–406 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mishra, S., Raman, V., Saurabh, S., Sikdar, S.: König deletion sets and vertex covers above the matching size. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 836–847. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  21. Pilipczuk, M., Pilipczuk, M.: Finding a maximum induced degenerate subgraph faster than 2 n. In: Thilikos, D.M., Woeginger, G.J. (eds.) IPEC 2012. LNCS, vol. 7535, pp. 3–12. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  22. Razgon, I.: Exact computation of maximum induced forest. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 160–171. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  23. Razgon, I.: Computing Minimum Directed Feedback Vertex Set in O *(1.9977n). In: ICTCS, pp. 70–81 (2007)

    Google Scholar 

  24. Robson, J.M.: Algorithms for maximum independent sets. J. Algorithms 7(3), 425–440 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yannakakis, M., Gavril, F.: The maximum k-colorable subgraph problem for chordal graphs. Information Processing Letters 24(2), 133–137 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. University of Maryland, USA

    Rajesh Chitnis

  2. University of Bergen, Norway

    Fedor V. Fomin, Daniel Lokshtanov & Saket Saurabh

  3. Institute of Mathematical Sciences, India

    Pranabendu Misra, M. S. Ramanujan & Saket Saurabh

Authors
  1. Rajesh Chitnis
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  2. Fedor V. Fomin
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  3. Daniel Lokshtanov
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  4. Pranabendu Misra
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  5. M. S. Ramanujan
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  6. Saket Saurabh
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Editor information

Editors and Affiliations

  1. Royal Holloway, University of London, Egham Hill, TW20 0EX, Egham, UK

    Gregory Gutin

  2. Institute of Information Systems, Vienna University of Technology, (184/3), Favoritenstraße 9-11, 1040, Vienna, Austria

    Stefan Szeider

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Chitnis, R., Fomin, F.V., Lokshtanov, D., Misra, P., Ramanujan, M.S., Saurabh, S. (2013). Faster Exact Algorithms for Some Terminal Set Problems. In: Gutin, G., Szeider, S. (eds) Parameterized and Exact Computation. IPEC 2013. Lecture Notes in Computer Science, vol 8246. Springer, Cham. https://doi.org/10.1007/978-3-319-03898-8_14

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  • DOI: https://doi.org/10.1007/978-3-319-03898-8_14

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-03897-1

  • Online ISBN: 978-3-319-03898-8

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