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Complexity of Finding Maximum Regular Induced Subgraphs with Prescribed Degree

  • Yuichi Asahiro
  • Hiroshi Eto
  • Takehiro Ito
  • Eiji Miyano
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8070)

Abstract

We study the problem of finding a maximum vertex-subset S of a given graph G such that the subgraph G[S] induced by S is r-regular for a prescribed degree r ≥ 0. We also consider a variant of the problem which requires G[S] to be r-regular and connected. Both problems are known to be NP-hard even to approximate for a fixed constant r. In this paper, we thus consider the problems whose input graphs are restricted to some special classes of graphs. We first show that the problems are still NP-hard to approximate even if r is a fixed constant and the input graph is either bipartite or planar. On the other hand, both problems are tractable for graphs having tree-like structures, as follows. We give linear-time algorithms to solve the problems for graphs with bounded treewidth; we note that the hidden constant factor of our running time is just a single exponential of the treewidth. Furthermore, both problems are solvable in polynomial time for chordal graphs.

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References

  1. 1.
    Asahiro, Y., Eto, H., Miyano, E.: Inapproximability of maximum r-regular induced connected subgraph problems. IEICE Transactions on Information and Systems E96-D, 443–449 (2013)Google Scholar
  2. 2.
    Betzler, N., Niedermeier, R., Uhlmann, J.: Tree decompositions of graphs: saving memory in dynamic programming. Discrete Optimization 3, 220–229 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Blair, J.R.S., Peyton, B.: An introduction to chordal graphs and clique trees. Graph Theory and Sparse Matrix Computation 56, 1–29 (1993)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Computing 25, 1305–1317 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brandstädg, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM (1999)Google Scholar
  6. 6.
    Cameron, K.: Induced matchings. Discrete Applied Mathematics 24, 97–102 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Cardoso, D.M., Kamiński, M., Lozin, V.: Maximum k-regular induced subgraphs. J. Combinatorial Optimization 14, 455–463 (2007)zbMATHCrossRefGoogle Scholar
  8. 8.
    Courcelle, B.: Graph rewriting: an algebraic and logic approach, Handbook of Theoretical Computer Science, vol. B, pp. 193–242. MIT Press (1990)Google Scholar
  9. 9.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  10. 10.
    Gavril, F.: Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM J. Computing 1, 180–187 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Håstad, J.: Clique is hard to approximate within n 1 − ε. Acta Mathematica 182, 105–142 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Kann, V.: Strong lower bounds on the approximability of some NPO PB-complete maximization problems. In: Hájek, P., Wiedermann, J. (eds.) MFCS 1995. LNCS, vol. 969, pp. 227–236. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  13. 13.
    Lampis, M.: Algorithmic meta-theorems for restrictions of treewidth. Algorithmica 64, 19–37 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Lund, C., Yannakakis, M.: The approximation of maximum subgraph problems. In: Lingas, A., Carlsson, S., Karlsson, R. (eds.) ICALP 1993. LNCS, vol. 700, pp. 40–51. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  15. 15.
    Orlovich, Y., Finke, G., Gordon, V., Zverovich, I.: Approximability results for the maximum and minimum maximal induced matching problems. Discrete Optimization 5, 584–593 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Stewart, I.A.: Deciding whether a planar graph has a cubic subgraph is NP-complete. Discrete Mathematics 126, 349–357 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Stewart, I.A.: Finding regular subgraphs in both arbitrary and planar graphs. Discrete Applied Mathematics 68, 223–235 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Stewart, I.A.: On locating cubic subgraphs in bounded-degree connected bipartite graphs. Discrete Mathematics 163, 319–324 (1997)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Yuichi Asahiro
    • 1
  • Hiroshi Eto
    • 2
  • Takehiro Ito
    • 3
  • Eiji Miyano
    • 2
  1. 1.Department of Information ScienceKyushu Sangyo UniversityJapan
  2. 2.Department of Systems Design and InformaticsKyushu Institute of TechnologyJapan
  3. 3.Graduate School of Information SciencesTohoku UniversityJapan

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