A Computational Proof of Complexity of Some Restricted Counting Problems

  • Authors:

  • Jin-Yi Cai;Pinyan Lu;Mingji Xia

  • Affiliations:

  • Computer Sciences Department, University of Wisconsin, Madison, USA WI 53706;Microsoft Research Asia, Beijing, P.R. China 100190;Computer Sciences Department, University of Wisconsin, Madison, USA WI 53706 and State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, P.R. China 1 ...

  • Venue:
  • TAMC '09 Proceedings of the 6th Annual Conference on Theory and Applications of Models of Computation
  • Year:
  • 2009

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Abstract

We explore a computational approach to proving intractability of certain counting problems. More specifically we study the complexity of Holant of 3-regular graphs. These problems include concrete problems such as counting the number of vertex covers or independent sets for 3-regular graphs. The high level principle of our approach is algebraic, which provides sufficient conditions for interpolation to succeed. Another algebraic component is holographic reductions . We then analyze in detail polynomial maps on ***2 induced by some combinatorial constructions. These maps define sufficiently complicated dynamics of ***2 that we can only analyze them computationally. We use both numerical computation (as intuitive guidance) and symbolic computation (as proof theoretic verification) to derive that a certain collection of combinatorial constructions, in myriad combinations, fulfills the algebraic requirements of proving #P-hardness. The final result is a dichotomy theorem for a class of counting problems.