# 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…

## 29 Citations

Computing the partition function for graph homomorphisms with multiplicities

- Mathematics, Computer ScienceJ. Comb. Theory, Ser. A
- 2016

Computing the Partition Function for Cliques in a Graph

- Mathematics, Computer ScienceTheory Comput.
- 2015

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.

Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials

- Computer Science, MathematicsElectron. Notes Discret. Math.
- 2017

Zero-Free Regions of Partition Functions with Applications to Algorithms and Graph Limits

- Mathematics, Computer ScienceComb.
- 2018

A class of edge-coloring models whose partition functions do not evaluate to zero on bounded degree graphs is identified and a quasi-polynomial time approximation scheme for computing these partition functions is given.

A Deterministic Algorithm for Counting Colorings with 2-Delta Colors

- Mathematics, Computer Science2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
- 2019

This work gives a polynomial time deterministic approximation algorithm (an FPTAS) for counting the number of q-colorings of a graph of maximum degree Delta, provided only that q ≥ 2Delta, and matches the natural bound for randomized algorithms obtained by a straightforward application of Markov chain Monte Carlo.

A dichotomy for bounded degree graph homomorphisms with nonnegative weights

- MathematicsICALP
- 2020

It is proved that for nonnegative symmetric $A$, either $Z_A(G)$ is in polynomial time for all graphs £G, or it is #P-hard for bounded degree (and simple) graphs $G$.

Dichotomy for Graph Homomorphisms with Complex Values on Bounded Degree Graphs

- Mathematics2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
- 2020

It is proved that either Garrow Z_{\mathrm{A}}(G)$ is computable in polynomial-time for every $G$, or for some $\Delta > 0$ it is #P-hard over (simple) graphs $G$ with maximum degree $\Delta (G)\leq\Delta$.

A deterministic algorithm for counting colorings with 2Δ colors

- Mathematics, Computer ScienceArXiv
- 2019

A polynomial time deterministic approximation algorithm (an FPTAS) for counting the number of q-colorings of a graph of maximum degree Δ, provided only that q ≥ 2Δ, and matches the natural bound for randomized algorithms obtained by a straightforward application of Markov chain Monte Carlo.

Approximation Algorithms for Complex-Valued Ising Models on Bounded Degree Graphs

- Computer Science, MathematicsQuantum
- 2019

A deterministic polynomial-time approximation scheme for the Ising model partition function when the interactions and external fields are absolutely bounded close to zero and it is proved that for this class of Ising models the partition function does not vanish.

A short proof of the equivalence of left and right convergence for sparse graphs

- MathematicsEur. J. Comb.
- 2016

## References

SHOWING 1-10 OF 17 REFERENCES

Computing the Partition Function for Cliques in a Graph

- Mathematics, Computer ScienceTheory Comput.
- 2015

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.

Homomorphisms of Edge-Colored Graphs and Coxeter Groups

- Mathematics
- 1998

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

- Computer Science, MathematicsSIAM J. Comput.
- 2013

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

- MathematicsSTOC '06
- 2006

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

- MathematicsColloquium Publications
- 2012

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.

Linear degree extractors and the inapproximability of max clique and chromatic number

- Computer Science, MathematicsSTOC '06
- 2006

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

- MathematicsJACM
- 1998

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

- Computer Science, MathematicsProceedings of Computational Complexity (Formerly Structure in Complexity Theory)
- 1996

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.