Counting Homomorphisms to K4-minor-free Graphs, modulo 2

We study the problem of computing the parity of the number of homomorphisms from an input graph $G$ to a fixed graph $H$. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph $H$ and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class $\oplus\mathrm{P}$ of parity problems. We verify their conjecture for all graphs $H$ that exclude the complete graph on $4$ vertices as a minor. Further, we rule out the existence of a subexponential-time algorithm for the $\oplus\mathrm{P}$-complete cases, assuming the randomised Exponential Time Hypothesis. Our proofs introduce a novel method of deriving hardness from globally defined substructures of the fixed graph $H$. Using this, we subsume all prior progress towards resolving the conjecture (Faben and Jerrum [ToC'15]; Gobel, Goldberg and Richerby [ToCT'14,'16]). As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most $3$, as well as a full classification for the problem of counting list homomorphisms, modulo $2$.

[1]  Ian Parberry,et al.  On the Construction of Parallel Computers from Various Bases of Boolean Functions , 1986, Theor. Comput. Sci..

[2]  Leslie Ann Goldberg,et al.  Approximately Counting H-Colourings is #BIS-Hard , 2015, ArXiv.

[3]  Dániel Marx,et al.  Homomorphisms are a good basis for counting small subgraphs , 2017, STOC.

[4]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[5]  John Faben.  The complexity of counting solutions to Generalised Satisfiability Problems modulo k , 2008, ArXiv.

[6]  Marc Roth,et al.  Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory , 2019, SODA.

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

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

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

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

[11]  K. Wagner,et al.  Graph Minor Theory , 2005 .

[12]  Leslie Ann Goldberg,et al.  Approximately Counting H-Colorings is $\#\mathrm{BIS}$-Hard , 2016, SIAM J. Comput..

[13]  Thomas Schwentick,et al.  When is the evaluation of conjunctive queries tractable? , 2001, STOC '01.

[14]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[15]  Martin Grohe.  The complexity of homomorphism and constraint satisfaction problems seen from the other side , 2007, JACM.

[16]  Seinosuke Toda,et al.  PP is as Hard as the Polynomial-Time Hierarchy , 1991, SIAM J. Comput..

[17]  Radu Curticapean,et al.  The Exponential-Time Complexity of Counting (Quantum) Graph Homomorphisms , 2019, WG.

[18]  Russell Impagliazzo,et al.  Complexity of k-SAT , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[19]  Dániel Marx,et al.  Exponential Time Complexity of the Permanent and the Tutte Polynomial , 2010, TALG.

[20]  Martin E. Dyer,et al.  On the relative complexity of approximate counting problems , 2000, APPROX.

[21]  Martin E. Dyer,et al.  The Relative Complexity of Approximate Counting Problems , 2000, Algorithmica.

[22]  Jörg Flum,et al.  The Parameterized Complexity of Counting Problems , 2004, SIAM J. Comput..

[23]  Pavol Hell,et al.  List Homomorphisms to Reflexive Graphs , 1998, J. Comb. Theory, Ser. B.

[24]  Martin Charles Golumbic,et al.  Perfect Elimination and Chordal Bipartite Graphs , 1978, J. Graph Theory.

[25]  Leslie Ann Goldberg,et al.  The Complexity of Approximately Counting Tree Homomorphisms , 2013, TOCT.

[26]  Maria J. Serna,et al.  Counting H-Colorings of Partial k-Trees , 2001, COCOON.

[27]  Steven M. Kelk.  On the relative complexity of approximately counting H-colourings , 2003 .

[28]  R. Duffin.  Topology of series-parallel networks , 1965 .

[29]  Phokion G. Kolaitis,et al.  Conjunctive-query containment and constraint satisfaction , 1998, PODS.

[30]  Graham R. Brightwell,et al.  Graph Homomorphisms and Phase Transitions , 1999, J. Comb. Theory, Ser. B.

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

[32]  V. Sós,et al.  Counting Graph Homomorphisms , 2006 .

[33]  Christos H. Papadimitriou,et al.  Two remarks on the power of counting , 1983, Theoretical Computer Science.

[34]  Leslie Ann Goldberg,et al.  A Complexity Trichotomy for Approximately Counting List H-Colorings , 2017, ACM Trans. Comput. Theory.

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

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