Computing the partition function for graph homomorphisms

@article{Barvinok2017ComputingTP,
  title={Computing the partition function for graph homomorphisms},
  author={Alexander I. Barvinok and Pablo Sober{\'o}n},
  journal={Combinatorica},
  year={2017},
  volume={37},
  pages={633-650}
}
We introduce the partition function of edge-colored graph homomorphisms, of which the usual partition function of graph homomorphisms is a specialization, and present an efficient algorithm to approximate it in a certain domain. Corollaries include effcient algorithms for computing weighted sums approximating the number of k-colorings and the number of independent sets in a graph, as well as an effcient procedure to distinguish pairs of edge-colored graphs with many color-preserving… 
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References

SHOWING 1-10 OF 17 REFERENCES
Computing the Partition Function for Cliques in a Graph
  • A. Barvinok
  • Mathematics, Computer Science
    Theory Comput.
  • 2015
TLDR
A deterministic algorithm which, given a graph G with n vertices and an integer 1 0 is an absolute constant, allows us to tell apart the graphs that do not have m-subsets of high density from the graph that have sufficiently many m- Subsets ofhigh density.
The complexity of partition functions
Homomorphisms of Edge-Colored Graphs and Coxeter Groups
AbstractLet $$G_1 = (V_1 ,E_1 ){\text{ and }}G_2 = (V_2 ,E_2 )$$ be two edge-colored graphs (without multiple edges or loops). A homomorphism is a mappingϕ : $$V_1 \mapsto V_2 $$ for which, for…
Graph Homomorphisms with Complex Values: A Dichotomy Theorem
TLDR
This paper proves a complete dichotomy theorem for this problem, showing that $Z_{\mathbf{A}} (\cdot)$ is either computable in polynomial-time or \#P-hard, depending explicitly on the matrix $\mathBF{A}$.
Counting independent sets up to the tree threshold
TLDR
It is shown that on any graph of maximum degree Δ correlations decay with distance at least as fast as they do on the regular tree of the same degree, which resolves an open conjecture in statistical physics.
Large Networks and Graph Limits
TLDR
Laszlo Lovasz has written an admirable treatise on the exciting new theory of graph limits and graph homomorphisms, an area of great importance in the study of large networks.
Coloring k-colorable graphs using relatively small palettes
Linear degree extractors and the inapproximability of max clique and chromatic number
TLDR
New extractors which require only log n + O(1) additional random bits for sources with constant entropy rate are constructed, and dispersers, which are similar to one-sided extractors, are built, which use an arbitrarily small constant times log n additional Randomness Extractor to within n1-ε are NP-hard.
Approximate graph coloring by semidefinite programming
TLDR
A duality relationship established between the value of the optimum solution to the authors' semidefinite program and the Lovász &thgr;-function is established and lower bounds on the gap between the best known approximation ratio in terms of n are shown.
Zero knowledge and the chromatic number
  • U. Feige, J. Kilian
  • Computer Science, Mathematics
    Proceedings of Computational Complexity (Formerly Structure in Complexity Theory)
  • 1996
TLDR
A new technique, inspired by zero-knowledge proof systems, is presented for proving lower bounds on approximating the chromatic number of a graph, and the result matches (up to low order terms) the known gap for approximation the size of the largest independent set.
...
1
2
...