Complexity classification of counting graph homomorphisms modulo a prime number

Counting graph homomorphisms and its generalizations such as the Counting Constraint Satisfaction Problem (CSP), its variations, and counting problems in general have been intensively studied since the pioneering work of Valiant. While the complexity of exact counting of graph homomorphisms (Dyer and Greenhill, 2000) and the counting CSP (Bulatov, 2013, and Dyer and Richerby, 2013) is well understood, counting modulo some natural number has attracted considerable interest as well. In their 2015 paper Faben and Jerrum suggested a conjecture stating that counting homomorphisms to a fixed graph H modulo a prime number is hard whenever it is hard to count exactly, unless H has automorphisms of certain kind. In this paper we confirm this conjecture. As a part of this investigation we develop techniques that widen the spectrum of reductions available for modular counting and apply to the general CSP rather than being limited to graph homomorphisms.

[1]  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..

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

[3]  Mark Jerrum,et al.  Polynomial-Time Approximation Algorithms for the Ising Model , 1990, SIAM J. Comput..

[4]  Andrei A. Bulatov,et al.  The complexity of the counting constraint satisfaction problem , 2008, JACM.

[5]  Nadia Creignou,et al.  Complexity of Generalized Satisfiability Counting Problems , 1996, Inf. Comput..

[6]  W. Imrich,et al.  Handbook of Product Graphs, Second Edition , 2011 .

[7]  Leslie Ann Goldberg,et al.  The complexity of counting homomorphisms to cactus graphs modulo 2 , 2014, TOCT.

[8]  Alexander I. Barvinok,et al.  Combinatorics and Complexity of Partition Functions , 2017, Algorithms and combinatorics.

[9]  Tobias Friedrich,et al.  On Counting (Quantum-)Graph Homomorphisms in Finite Fields , 2021, ICALP.

[10]  Martin E. Dyer,et al.  An Effective Dichotomy for the Counting Constraint Satisfaction Problem , 2010, SIAM J. Comput..

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

[12]  Andreas Göbel,et al.  Counting Homomorphisms to Trees Modulo a Prime , 2018, MFCS.

[13]  Libor Barto,et al.  Polymorphisms, and How to Use Them , 2017, The Constraint Satisfaction Problem.

[14]  Leslie Ann Goldberg,et al.  Counting Homomorphisms to Square-Free Graphs, Modulo 2 , 2016, TOCT.

[15]  Peter Jeavons,et al.  On the Algebraic Structure of Combinatorial Problems , 1998, Theor. Comput. Sci..

[16]  Leslie Ann Goldberg,et al.  Counting Homomorphisms to K4-Minor-Free Graphs, Modulo 2 , 2021, SIAM J. Discret. Math..

[17]  Ulrich Hertrampf,et al.  Relations Among Mod-Classes , 1990, Theor. Comput. Sci..

[18]  Andrei A. Bulatov,et al.  Towards a dichotomy theorem for the counting constraint satisfaction problem , 2007, Inf. Comput..

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

[20]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[21]  Peter Jeavons,et al.  Classifying the Complexity of Constraints Using Finite Algebras , 2005, SIAM J. Comput..

[22]  Jin-Yi Cai,et al.  On the Power of Parity Polynomial Time , 1989, STACS.

[23]  Andrei A. Bulatov,et al.  Counting Homomorphisms Modulo a Prime Number , 2019, MFCS.

[24]  Elliott H. Lieb,et al.  A general Lee-Yang theorem for one-component and multicomponent ferromagnets , 1981 .

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

[26]  Marc Gyssens,et al.  How to Determine the Expressive Power of Constraints , 1999, Constraints.

[27]  Leslie Ann Goldberg,et al.  Counting Homomorphisms to K4-minor-free Graphs, modulo 2 , 2021, SODA.

[28]  Phokion G. Kolaitis.  Constraint Satisfaction, Complexity, and Logic , 2004, SETN.

[29]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[30]  Mark Jerrum,et al.  The Complexity of Parity Graph Homomorphism: An Initial Investigation , 2013, Theory Comput..