The complexity of counting graph homomorphisms

@article{Dyer2000TheCO,
  title={The complexity of counting graph homomorphisms},
  author={M. Dyer and Catherine S. Greenhill},
  journal={Random Struct. Algorithms},
  year={2000},
  volume={17},
  pages={260-289}
}
The problem of counting homomorphisms from a general graph G to a fixed graph H is a natural generalisation of graph colouring, with important applications in statistical physics. The problem of deciding whether any homomorphism exists was considered by Hell and Nešetřil. They showed that decision is NPcomplete unless H has a loop or is bipartite; otherwise it is in P. We consider the problem of exactly counting such homomorphisms, and give a similarly complete characterisation. We show that… Expand
The complexity of counting graph homomorphisms (extended abstract)
TLDR
The theorems provide the first proof of #P-completeness of the partition function of certain models from statistical physics, such as the Widom-Rowlinson model, even in graphs of maximum degree 3. Expand
Counting, modular counting and graph homomorphisms
TLDR
This thesis studies the complexity of various problems related to graph homomorphisms, and gives an explicit characterisation of the dichotomy theorem — counting list M -partitions is tractable (in FP) if the matrix M has a structure called a derectangularising sequence. Expand
Counting Homomorphisms Modulo a Prime Number
TLDR
It is extended to show that the #_p GraphHom(H) problem is#_p P-hard whenever the derived graph associated with H is square-free and is not a star, which completely classifies the complexity of #_ p GraphHom (H) forsquare-free graphs H. Expand
The complexity of counting homomorphisms to cactus graphs modulo 2
TLDR
It is shown that, for some cactus graphs H, counting homomorphisms to H modulo 2 can be done in polynomial time and a dichotomy is given for the case in which H is a tree, which builds upon the work of Faben and Jerrum. Expand
Counting Homomorphisms to Cactus Graphs Modulo 2
TLDR
It is shown that, for some cactus graphs H, counting homomorphisms to H modulo 2 can be done in polynomial time and the result builds upon the work of Faben and Jerrum, who gave a dichotomy for the case in which H is a tree. Expand
Counting Homomorphisms to Cactus Graphs Modulo 2ú
Abstract A homomorphism from a graph G to a graph H is a function from V (G) to V (H) that preserves edges. Many combinatorial structures that arise in mathematics and computer science can beExpand
Counting Homomorphisms to Square-Free Graphs, Modulo 2
TLDR
The following dichotomy is shown: for any H that contains no 4-cycles, ⊕HomsToH is either in polynomial time or is⊕P-complete, which partially confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of tree-width-2 graphs called cactus graphs. Expand
THE COMPLEXITY OF COUNTING SURJECTIVE
4 A homomorphism from a graph G to a graph H is a function from the vertices of G to the 5 vertices of H that preserves edges. A homomorphism is surjective if it uses all of the vertices 6 of H andExpand
The Complexity of Counting Surjective Homomorphisms and Compactions
TLDR
A complete characterisation of the complexity of counting surjective homomorphisms from an input graph G to a fixed graph H is given and a dichotomy is pointed out for the complexityof the respective approximate counting problems (in the connected case). Expand
Counting Partitions of Graphs
TLDR
It turns out that, among matrices not acccounted for by the existing results on counting homomorphisms, all matrices which do not contain the matrices for independent sets or cliques yield tractable counting problems. Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 30 REFERENCES
On Approximately Counting Colorings of Small Degree Graphs
TLDR
A computer-assisted proof technique is used to establish rapid mixing of a new "heat bath" Markov chain on colorings using the method of path coupling and gives a general proof that the problem of exactly counting the number of proper k-colorings of graphs with maximum degree $\Delta$ is complete. Expand
On the complexity of H-coloring
TLDR
The natural conjecture, formulated in several sources, asserts that the H-coloring problem is NP-complete for any non-bipartite graph H, and a proof of this conjecture is given. Expand
Homomorphisms of 3-chromatic graphs
TLDR
The principal result (Theorem 2) provides necessary conditions for the existence of a homomorphism onto a prescribed target and it is shown that iterated cartesian products of the Petersen graph form an infinite family of vertex transitive graphs no one of which is the homomorphic image of any other. Expand
The Complexity of Counting in Sparse, Regular, and Planar Graphs
  • S. Vadhan
  • Computer Science, Mathematics
  • SIAM J. Comput.
  • 2001
TLDR
It is proved that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree. Expand
On unavoidable digraphs in orientations of graphs
TLDR
It is proved that among all G for which G D, the minimum chromatic number is equal to the minimum k for which Kk hom(D) is the set of homomorphs of D, and necessary and sufficient conditions are given for a directed graph to have a homomorphism into a given transitive tournament, directed path, or directed cycle. Expand
NP-completeness of a family of graph-colouring problems
TLDR
Each of these problems is shown to be NP-complete by constructing a polynomial transformation from 3-satisfiability to (2 r +1, r )-colourability, valid for each value of r. Expand
Graph Homomorphisms and Phase Transitions Bell Laboratories 2c-379 Lucent Technologies 700 Mountain Ave
We model physical systems with \hard constraints" by the space Hom(G;H) of homomorphisms from a locally nite graph G to a xed nite constraint graph H. For any assignment of positive real activitiesExpand
The Complexity of Planar Counting Problems
TLDR
It is proved that there are no $\epsilon$-approximation algorithms for the problems of maximizing or minimizing a linear objective function subject to a planar system of linear inequality constraints over the integers. Expand
On Markov Chains for Independent Sets
TLDR
A new rapidly mixing Markov chain for independent sets is defined and a polynomial upper bound for the mixing time of the new chain is obtained for a certain range of values of the parameter ?, which is wider than the range for which the mixingTime of the Luby?Vigoda chain is known to be polynomially bounded. Expand
The complexity of counting colourings and independent sets in sparse graphs and hypergraphs
TLDR
Using polynomial interpolation techniques, it is shown that certain counting problems involving colourings of graphs and independent sets in hypergraphs are #P-complete and efficient approximate counting is the most one can realistically expect to achieve. Expand
...
1
2
3
...