Graph Homomorphisms with Complex Values: A Dichotomy Theorem

Graph homomorphism problem has been studied intensively. Given an m × m symmetric matrix A, the graph homomorphism function ZA(G) is defined as ZA(G) = σξ:V →[m] (u,v)∈E ΠAξ(u),ξ(v), where G = (V,E) is any undirected graph. The function ZA(G) can encode many interesting graph properties, including counting vertex covers and k-colorings. We study the computational complexity of ZA(G), for arbitrary complex valued symmetric matrices A. Building on the work by Dyer and Greenhill [1], Bulatov and Grohe [2], and especially the recent beautiful work by Goldberg, Grohe, Jerrum and Thurley [3], we prove a complete dichotomy theorem for this problem.

[1]  L. Lovasz,et al.  Reflection positivity, rank connectivity, and homomorphism of graphs , 2004, math/0404468.

[2]  L. Lovász.  Operations with structures , 1967 .

[3]  Jin-Yi Cai,et al.  Holant problems and counting CSP , 2009, STOC '09.

[4]  H. Niederreiter,et al.  Finite Fields: Encyclopedia of Mathematics and Its Applications. , 1997 .

[5]  公庄 庸三.  Basic Algebra = 代数学入門 , 2002 .

[6]  Marc Thurley.  The Complexity of Partition Functions on Hermitian Matrices , 2010, ArXiv.

[7]  L. Carlitz.  Kloosterman sums and finite field extensions , 1969 .

[8]  Jaroslav Nesetril,et al.  Graphs and homomorphisms , 2004, Oxford lecture series in mathematics and its applications.

[9]  Jaroslav Nesetril,et al.  On the complexity of H-coloring , 1990, J. Comb. Theory, Ser. B.

[10]  Jin-Yi Cai,et al.  A Decidable Dichotomy Theorem on Directed Graph Homomorphisms with Non-negative Weights , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[11]  Martin E. Dyer,et al.  On the complexity of #CSP , 2010, STOC '10.

[12]  Leslie Ann Goldberg,et al.  A Complexity Dichotomy for Partition Functions with Mixed Signs , 2008, SIAM J. Comput..

[13]  Martin E. Dyer,et al.  On counting homomorphisms to directed acyclic graphs , 2006, JACM.

[14]  R. Tennant.  Algebra , 1941, Nature.

[15]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[16]  Guoqiang Ge.  Testing equalities of multiplicative representations in polynomial time , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[17]  László Lovász,et al.  The rank of connection matrices and the dimension of graph algebras , 2004, Eur. J. Comb..

[18]  Alexander Schrijver,et al.  Graph invariants in the spin model , 2009, J. Comb. Theory, Ser. B.

[19]  R. Leighton,et al.  Feynman Lectures on Physics , 1971 .

[20]  R. Feynman,et al.  The Feynman Lectures on Physics Addison-Wesley Reading , 1963 .

[21]  K. Conrad,et al.  Finite Fields , 2018, Series and Products in the Development of Mathematics.

[22]  Jin-Yi Cai,et al.  Holographic algorithms: from art to science , 2007, STOC '07.

[23]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[24]  Martin E. Dyer,et al.  The complexity of counting graph homomorphisms , 2000, Random Struct. Algorithms.

[25]  N. Jacobson,et al.  Basic Algebra II , 1989 .

[26]  Jin-Yi Cai,et al.  Non-negatively Weighted #CSP: An Effective Complexity Dichotomy , 2010, 2011 IEEE 26th Annual Conference on Computational Complexity.

[27]  Leslie G. Valiant,et al.  Holographic Algorithms (Extended Abstract) , 2004, FOCS.

[28]  H. Lenstra,et al.  Algorithms in algebraic number theory , 1992, math/9204234.

[29]  Patrick J. Morandi.  Field and Galois theory , 1996 .

[30]  Andrei A. Bulatov,et al.  The complexity of partition functions , 2005, Theor. Comput. Sci..

[31]  Martin E. Dyer,et al.  The complexity of weighted and unweighted #CSP , 2010, J. Comput. Syst. Sci..

[32]  Sanjeev Khanna,et al.  Complexity classifications of Boolean constraint satisfaction problems , 2001, SIAM monographs on discrete mathematics and applications.

[33]  Rudolf Lide,et al.  Finite fields , 1983 .

[34]  Ker-I Ko,et al.  Complexity Theory of Real Functions , 1991, Progress in Theoretical Computer Science.

[35]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..