#3-Regular Bipartite Planar Vertex Cover is #P-Complete

  • Mingji Xia
  • Wenbo Zhao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3959)


We generalize the polynomial interpolation method by giving a sufficient condition, which guarantees that the coefficients of a polynomial are uniquely determined by its values on a recurrence sequence. Using this method, we show that #3-Regular Bipartite Planar Vertex Cover is #P-complete. The result is unexpected, since the same question for 2-regular graph is in FP.


Polynomial Time Vertex Cover Main Lemma Counting Problem Linear Equation System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Hunt III, H.B., Marathe, M.V., Radhakrishnan, V., Stearns, R.E.: The Complexity of Planar Counting Problems. SIAM J. Comput. 27, 1142–1167 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  2. Liskiewicz, M., Ogihara, M., Toda, S.: The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes. Theor. Comput. Sci. 304, 129–156 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  3. Greenhill, C.S.: The complexity of counting colourings and independent sets in sparse graphs and hypergraphs. Computational Complexity 9, 52–72 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  4. Papadimitriou, C.: Computational Complexity. Addison–Wesley, Reading (1994)zbMATHGoogle Scholar
  5. Vadhan, S.P.: The Complexity of Counting in Sparse, Regular, and Planar Graphs. SIAM Journal on Computing 31, 398–427 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  6. Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM Journal on Computing 8, 410–421 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  7. Valiant, L.G.: Holographic Algorithms (Extended Abstract). In: FOCS, pp. 306–315 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Mingji Xia
    • 1
  • Wenbo Zhao
    • 2
  1. 1.Institute of SoftwareChinese Academy of SciencesBeijingChina
  2. 2.Graduate University of Chinese Academy of SciencesBeijingChina

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