A Computational Framework for the Study of Partition Functions and Graph Polynomials

Partition functions and graph polynomials have found many applications in combinatorics, physics, biology and even the mathematics of finance. Studying their complexity poses some problems. To capture the complexity of their combinatorial nature, the Turing model of computation and Valiant's notion of counting complexity classes seem most natural. To capture the algebraic and numeric nature of partition functions as real or complex valued functions, the Blum-Shub-Smale (BSS) model of computation seems more natural. As a result many papers use a naive hybrid approach in discussing their complexity or restrict their considerations to sub-fields of C which can be coded in a way to allow dealing with Turing computability. In this paper we propose a unified natural framework for the study of computability and complexity of partition functions and graph polynomials and show how classical results can be cast in this framework.

[1]  Fan Chung Graham,et al.  On the Cover Polynomial of a Digraph , 1995, J. Comb. Theory, Ser. B.

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

[3]  Richard E. Ladner,et al.  On the Structure of Polynomial Time Reducibility , 1975, JACM.

[4]  Susan S. Margulies,et al.  Computer algebra, combinatorics, and complexity: hilbert's nullstellensatz and np-complete problems , 2008 .

[5]  Demetrios Achlioptas,et al.  The complexity of G-free colourability , 1997, Discret. Math..

[6]  Ilia Averbouch,et al.  The Complexity of Multivariate Matching Polynomials , 2007 .

[7]  Johann A. Makowsky,et al.  An extension of the bivariate chromatic polynomial , 2010, Eur. J. Comb..

[8]  D. Welsh,et al.  On the computational complexity of the Jones and Tutte polynomials , 1990, Mathematical Proceedings of the Cambridge Philosophical Society.

[9]  D. Bayer.  The division algorithm and the hilbert scheme , 1982 .

[10]  Klaus Meer.  Counting problems over the reals , 2000, Theor. Comput. Sci..

[11]  Felipe Cucker,et al.  Counting complexity classes for numeric computations II: algebraic and semialgebraic sets , 2003, STOC '04.

[12]  Béla Bollobás,et al.  The interlace polynomial of a graph , 2004, J. Comb. Theory, Ser. B.

[13]  Hermann A. Maurer,et al.  On Two-Symbol Complete E0L Forms , 1978, Theor. Comput. Sci..

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

[15]  Martin Grohe,et al.  Counting Homomorphisms and Partition Functions , 2009, AMS-ASL Joint Special Session.

[16]  Klaus Ambos-Spies.  On the Relative Complexity of Hard Problems for Complexity Classes without Complete Problems , 1989, Theor. Comput. Sci..

[17]  Saugata Basu.  A Complex Analogue of Toda’s Theorem , 2012, Found. Comput. Math..

[18]  Robert E. Tarjan,et al.  A Combinatorial Problem Which Is Complete in Polynomial Space , 1976, JACM.

[19]  Peter Tittmann,et al.  A new two-variable generalization of the chromatic polynomial , 2003, Discret. Math. Theor. Comput. Sci..

[20]  Bruno Courcelle,et al.  A Multivariate Interlace Polynomial and its Computation for Graphs of Bounded Clique-Width , 2008, Electron. J. Comb..

[21]  Felipe Cucker,et al.  Counting Complexity Classes for Numeric Computations. III: Complex Projective Sets , 2005, Found. Comput. Math..

[22]  Christian Hoffmann,et al.  A Most General Edge Elimination Polynomial - Thickening of Edges , 2008, Fundam. Informaticae.

[23]  Johann A. Makowsky,et al.  The enumeration of vertex induced subgraphs with respect to the number of components , 2008, Eur. J. Comb..

[24]  Markus Bläser,et al.  On the Complexity of the Interlace Polynomial , 2007, STACS.

[25]  Béla Bollobás,et al.  A Tutte Polynomial for Coloured Graphs , 1999, Combinatorics, Probability and Computing.

[26]  W. Marsden.  I and J , 2012 .

[27]  Robin Wilson,et al.  Modern Graph Theory , 2013 .

[28]  G. Birkhoff.  A Determinant Formula for the Number of Ways of Coloring a Map , 1912 .

[29]  Johann A. Makowsky,et al.  Complexity of the Bollobás–Riordan Polynomial. Exceptional Points and Uniform Reductions , 2008, Theory of Computing Systems.

[30]  Alan D. Sokal.  The multivariate Tutte polynomial (alias Potts model) for graphs and matroids , 2005, Surveys in Combinatorics.

[31]  W. T. Tutte,et al.  A Contribution to the Theory of Chromatic Polynomials , 1954, Canadian Journal of Mathematics.

[32]  Lorenzo Traldi,et al.  On the interlace polynomials , 2010, J. Comb. Theory, Ser. B.

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

[34]  Markus Bläser,et al.  Complexity of the Cover Polynomial , 2007, ICALP.

[35]  Saugata Basu,et al.  Polynomial Hierarchy, Betti Numbers, and a Real Analogue of Toda’s Theorem , 2008, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

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

[37]  Klaus Meer,et al.  On the Complexity of Combinatorial and Metafinite Generating Functions of Graph Properties in the Computational Model of Blum, Shub and Smale , 2000, CSL.

[38]  Christian Hoffmann,et al.  Computational complexity of graph polynomials , 2010 .

[39]  Mahmoud Fouz,et al.  Complexity and Approximability of the Cover Polynomial , 2011, computational complexity.

[40]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[41]  R. B. Potts.  Some generalized order-disorder transformations , 1952, Mathematical Proceedings of the Cambridge Philosophical Society.

[42]  Uwe Schöning.  A Uniform Approach to Obtain Diagonal Sets in Complexity Classes , 1982, Theor. Comput. Sci..

[43]  Ilia Averbouch.  Completeness and universality properties of graph invariants and graph polynomials , 2011 .

[44]  Shai Ben-David,et al.  A Note on Non-complete Problems in NPImage , 2000, J. Complex..

[45]  Béla Bollobás,et al.  The interlace polynomial: a new graph polynomial , 2000, SODA '00.

[46]  Jin-Yi Cai,et al.  Graph Homomorphisms with Complex Values: A Dichotomy Theorem , 2009, SIAM J. Comput..

[47]  Sagi Snir,et al.  Efficient approximation of convex recolorings , 2007, J. Comput. Syst. Sci..

[48]  Béla Bollobás,et al.  A Two-Variable Interlace Polynomial , 2004, Comb..

[49]  Johann A. Makowsky,et al.  On Counting Generalized Colorings , 2008, CSL.