Planarizing Gadgets for Perfect Matching Do Not Exist

  • Rohit Gurjar
  • Arpita Korwar
  • Jochen Messner
  • Simon Straub
  • Thomas Thierauf
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)

Abstract

To reduce a graph problem to its planar version, a standard technique is to replace crossings in a drawing of the input graph by planarizing gadgets. We show unconditionally that such a reduction is not possible for the perfect matching problem and also extend this to some other problems related to perfect matching. We further show that there is no planarizing gadget for the Hamiltonian cycle problem.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agrawal, M.: On derandomizing tests for certain polynomial identities. In: Proceedings of the Conference on Computational Complexity, pp. 355–359 (2003)Google Scholar
  2. 2.
  3. 3.
    Dahlhaus, E., Hajnal, P., Karpinski, M.: On the parallel complexity of Hamiltonian cycles and matching problem in dense graphs. J. Algorithms 15, 367–384 (1993)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Dahlhaus, E., Karpinski, M.: Matching and multidimensional matching in chordal and strongly chordal graphs. Disc. Appl. Math. 84, 79–91 (1998)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Datta, S., Kulkarni, R., Limaye, N., Mahajan, M.: Planarity, determinants, permanents, and (unique) matchings. ACM Trans. Comput. Theory, 10:1–10:20 (2010)Google Scholar
  6. 6.
    Datta, S., Kulkarni, R., Roy, S.: Deterministically isolating a perfect matching in bipartite planar graphs. Theor. Comput. Syst. 47, 737–757 (2010)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Datta, S., Kulkarni, R., Tewari, R.: Perfect matching in bipartite planar graphs is in UL. Technical Report TR10-201, ECCC (2011)Google Scholar
  8. 8.
    Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Gabow, H.N., Kaplan, H., Tarjan, R.E.: Unique maximum matching algorithms. J. Algorithms 40(2), 159–183 (2001)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Garey, M.R., Johnson, D.S., Tarjan, R.E.: The planar Hamiltonian circuit problem is NP-complete. SIAM J. Comput 5(4), 704–714 (1976)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gasarch, W.: Is there a nice gadget for showing planar HC is NPC? Computational Complexity Blog (2012), http://blog.computationalcomplexity.org/2012/01/is-there-nice-gadget-for-showing-planar.html
  13. 13.
    Hoang, T.M.: On the matching problem for special graph classes. In: Proceedings of the Conference on Computational Complexity, pp. 139–150 (2010)Google Scholar
  14. 14.
    Hoang, T.M., Mahajan, M., Thierauf, T.: On the Bipartite Unique Perfect Matching Problem. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 453–464. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Hopcroft, J., Karp, R.: An n 5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2, 225–231 (1973)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Karpinski, M., Rytter, W.: Fast Parallel Algorithms for Graph Matching Problems. Oxford University Press (1998)Google Scholar
  17. 17.
    Kasteleyn, P.W.: Graph theory and crystal physics. In: Harary, F. (ed.) Graph Theory and Theoret. Physics, pp. 43–110. Academic Press (1967)Google Scholar
  18. 18.
    Kozen, D.C., Vazirani, U.V., Vazirani, V.V.: NC Algorithms for Comparability Graphs, Interval Graphs, and Testing for Unique Perfect Matchings. In: Maheshwari, S.N. (ed.) FSTTCS 1985. LNCS, vol. 206, pp. 496–503. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  19. 19.
    Kozen, D.: The Design and Analysis of Algorithms. Springer (1991)Google Scholar
  20. 20.
    Kulkarni, R., Mahajan, M., Varadarajan, K.: Some perfect matchings and perfect half-integral matchings in NC. Chic. J. Theor. Comput. 2008(4) (2008)Google Scholar
  21. 21.
    Lev, G., Pippenger, M., Valiant, L.: A fast parallel algorithm for routing in permutation networks. IEEE Trans. Computers 30, 93–100 (1981)MathSciNetMATHGoogle Scholar
  22. 22.
    Lovasz, L., Plummer, M.D.: Matching theory. North-Holland (1986)Google Scholar
  23. 23.
    Mahajan, M., Varadarajan, K.R.: A new NC-algorithm for finding a perfect matching in bipartite planar and small genus graphs. In: 32th ACM Symp. Theo. Comput (STOC), pp. 351–357. ACM Press (2000)Google Scholar
  24. 24.
    Micali, S., Vazirani, V.: An \({O}(\sqrt{|v|}\cdot{|E|})\) algorithm for finding maximum matching in general graphs. In: FOCS, pp. 17–27 (1980)Google Scholar
  25. 25.
    Miller, G.L., Naor, J.S.: Flow in planar graphs with multiple sources and sinks. SIAM J. Comput. 24, 1002–1017 (1995)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Mulmuley, K., Vazirani, U., Vazirani, V.: Matching is as easy as matrix inversion. Combinatorica 7, 105–113 (1987)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Papadimitriou, C.H., Yannakakis, M.: The complexity of restricted spanning tree problems. J. ACM 29, 285–309 (1982)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Valiant, L.: The complexity of computing the permanent. Theoretical Computer Science 8, 189–201 (1979)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Vazirani, V.: NC algorithms for computing the number of perfect matchings in K 3,3-free graphs and related problems. Inf. Comput. 80, 152–164 (1989)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Vazirani, V.: A theory of alternating paths and blossoms for proving correctness of the \({O}(\sqrt{V}{E})\) general graph maximum matching algorithm. Combinatorica 14, 71–109 (1994)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Yuster, R.: Almost exact matchings. Algorithmica 63(1-2), 39–50 (2012)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Rohit Gurjar
    • 1
  • Arpita Korwar
    • 1
  • Jochen Messner
    • 2
  • Simon Straub
    • 3
  • Thomas Thierauf
    • 2
  1. 1.IIT KanpurIndia
  2. 2.Aalen UniversityGermany
  3. 3.University of UlmGermany

Personalised recommendations