Advertisement

Kernelization of Two Path Searching Problems on Split Graphs

  • Yongjie Yang
  • Yash Raj Shrestha
  • Wenjun Li
  • Jiong Guo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9711)

Abstract

In the k-Vertex-Disjoint Paths problem, we are given a graph G and k terminal pairs of vertices, and are asked whether there is a set of k vertex-disjoint paths linking these terminal pairs, respectively. In the k-Path problem, we are given a graph and are asked whether there is a path of length k. It is known that both problems are NP-hard even in split graphs, which are the graphs whose vertices can be partitioned into a clique and an independent set. We study kernelization for the two problems in split graphs. In particular, we derive a 4k vertex-kernel for the k-Vertex-Disjoint Paths problem and a \(\frac{3}{2}k^2+\frac{1}{2}k\) vertex-kernel for the k-Path problem.

References

  1. 1.
    Abu-Khzam, F.N.: A kernelization algorithm for \(d\)-hitting set. J. Comput. Syst. Sci. 76(7), 524–531 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abu-Khzam, F.N., Fellows, M.R., Langston, M.A., Suters, W.H.: Crown structures for vertex cover kernelization. Theor. Comput. Syst. 41(3), 411–430 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Adcock, A.B., Demaine, E.D., Demaine, M.L., O’Brien, M.P., Reidl, F., Villaamil, F.S., Sullivan, B.D.: Zig-zag numberlink is NP-complete. JIP 23(3), 239–245 (2015)Google Scholar
  4. 4.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci. 412(35), 4570–4578 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chor, B., Fellows, M., Juedes, D.W.: Linear kernels in linear time, or how to save k colors in \(O(n^2)\) steps. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 257–269. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    Frank, A.: Packing paths, circuits and cuts: a survey. In: Korte, B., Lovász, L., Prömel, H.J., Schrijver, A. (eds.) Paths, Flows, and VLSI-Layout, pp. 49–100. Springer, Berlin (1990)Google Scholar
  8. 8.
    Heggernes, P., Hof, P., van Leeuwen, E.J., Saei, R.: Finding disjoint paths in split graphs. Theor. Comput. Syst. 57(1), 140–159 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Heggernes, P., Lokshtanov, D., Mihai, R., Papadopoulos, C.: Cutwidth of split graphs and threshold graphs. SIAM J. Discrete Math. 25(3), 1418–1437 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jansen, B.M.P.: Turing kernelization for finding long paths and cycles in restricted graph classes. In: Schulz, A.S., Wagner, D. (eds.) ESA 2014. LNCS, vol. 8737, pp. 579–591. Springer, Heidelberg (2014)Google Scholar
  11. 11.
    Karp, R.M., Leighton, F.T., Rivest, R.L., Thompson, C.D., Vazirani, U.V., Vazirani, V.V.: Global wire routing in two-dimensional arrays. Algorithmica 2, 113–129 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kramer, M.R., van Leeuwen, J.: The complexity of wire-routing and finding minimum area layouts for arbitrary VLSI circuits. Adv. Comput. Res. 2, 129–146 (1984)Google Scholar
  13. 13.
    Lynch, J.F.: The equivalence of theorem proving and the interconnection problem. SIGDA Newsl. 5(3), 31–36 (1975)CrossRefGoogle Scholar
  14. 14.
    Merris, R.: Split graphs. Eur. J. Comb. 24(4), 413–430 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Müller, H.: Hamiltonian circuits in chordal bipartite graphs. Discrete Math. 156(1–3), 291–298 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Natarajan, S., Sprague, A.P.: Disjoint paths in circular arc graphs. Nord. J. Comput. 3(3), 256–270 (1996)MathSciNetGoogle Scholar
  17. 17.
    Prieto, E., Sloper, C.: Looking at the stars. In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 138–148. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Raman, V., Saurabh, S.: Short cycles make W-hard problems hard: FPT algorithms for W-hard problems in graphs with no short cycles. Algorithmica 52(2), 203–225 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Robertson, N., Seymour, P.D.: An outline of a disjoint path algorithm. In: Korte, B., Lovász, L., Prömel, H.J., Schrijver, A. (eds.) Paths, Flows, and VLSI-Layout, pp. 267–292. Springer, Berlin (1990)Google Scholar
  20. 20.
    Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theor. Ser. B 63(1), 65–110 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wang, J., Yang, Y., Guo, J., Chen, J.: Planar graph vertex partition for linear problem kernels. J. Comput. Syst. Sci. 79(5), 609–621 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yang, Y.: Towards optimal kernel for edge-disjoint triangle packing. Inf. Process. Lett. 114(7), 344–348 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Yongjie Yang
    • 1
  • Yash Raj Shrestha
    • 2
  • Wenjun Li
    • 3
  • Jiong Guo
    • 4
  1. 1.Universität des SaarlandesSaarbrückenGermany
  2. 2.Department of Management, Technology and EconomicsETH ZürichZürichSwitzerland
  3. 3.Hunan Provincial Key Laboratory of Intelligent Processing of Big Data on Transportation, School of Computer and Communication EngineeringChangsha University of Science and TechnologyChangshaChina
  4. 4.School of Computer Science and TechnologyShandong UniversityShandongChina

Personalised recommendations