# Clique , independent set , and graph coloring

@inproceedings{Pattillo2010CliqueI, title={Clique , independent set , and graph coloring}, author={Jeffrey Pattillo and S. Butenko}, year={2010} }

This article introduces the closely related maximum clique, maximum independent set, graph coloring, and minimum clique partitioning problems. The survey includes some of the most important results concerning these problems, including their computational complexity, known bounds, mathematical programming formulations, and exact and heuristic algorithms to solve them.

#### 8 Citations

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- Proceedings of the IEEE
- 2020

This article aims to illustrate the structural properties of integer programs that can be formulated as clique problems through multiple examples in communications and signal processing, mainly in maximum access for nonorthogonal multiple access networks, throughput maximization using index and instantly decodable network coding, collision-free radio-frequency identification networks, and resource allocation in cloud-radio access networks. Expand

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- 2016 American Control Conference (ACC)
- 2016

Lower bounds on the performance of greedy algorithms for submodular maximization, which depend on the clique number of the graph, are provided and graph-theoretic upper bounds in terms of the chromatic number ofThe graph are characterized. Expand

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- 2012

Abstract : This project focused on a multifaceted study of a class of cluster-detection problems arising in biological and social networks. This includes defining new cluster models and their… Expand

Distributed Submodular Maximization With Limited Information

- Computer Science
- IEEE Transactions on Control of Network Systems
- 2018

Lower bounds on the performance of greedy algorithms for submodular maximization, which depend on the clique number of a graph that captures the information structure, are provided and graph-theoretic upper bounds in terms of the chromatic number of the graph are characterized. Expand

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- 2017

We consider a class of distributed submodular maximization problems in which each agent must choose a single strategy from its strategy set. The global objective is to maximize a submodular function… Expand

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Using simulation, multiple classical performance metrics are derived to evaluate topology reconfiguration effectiveness for different point-to-point communication architecture attributes for the purpose of qualifying protocol design elements. Expand

#### References

SHOWING 1-10 OF 112 REFERENCES

The maximum clique problem

- Computer Science, Mathematics
- J. Glob. Optim.
- 1994

A survey of results concerning algorithms, complexity, and applications of the maximum clique problem is presented and enumerative and exact algorithms, heuristics, and a variety of other proposed methods are discussed. Expand

On cliques in graphs

- Mathematics
- 1965

A clique is a maximal complete subgraph of a graph. The maximum number of cliques possible in a graph withn nodes is determined. Also, bounds are obtained for the number of different sizes of cliques… Expand

A fast algorithm for the maximum clique problem

- Computer Science, Mathematics
- Discret. Appl. Math.
- 2002

A branch-and-bound algorithm for the maximum clique problem--which is computationally equivalent to the maximum independent (stable) set problem--is presented with the vertex order taken from a coloring of the vertices and with a new pruning strategy. Expand

Upper Bounds on the Order of a Clique of a Graph

- Mathematics
- 1972

This note presents an upper bound on the order of a largest complete subgraph (a clique) of a graph. Other upper bounds are given and the relationship between this bound and an existing bound is… Expand

An exact method for graph coloring

- Mathematics, Computer Science
- Comput. Oper. Res.
- 2006

These algorithms are the first to the knowledge to solve the COLOR02 instance 4-Inser_3 with an exact method and are useful to solve more quickly than other exact algorithms instances with small linearwidth, such as mug graphs. Expand

A Column Generation Approach for Graph Coloring

- Mathematics, Computer Science
- INFORMS J. Comput.
- 1996

This work uses a column generation method for implicit optimization of the linear program at each node of the branch-and-bound tree to solve small to moderate size problems quickly. Expand

A branch-and-cut algorithm for the maximum cardinality stable set problem

- Mathematics, Computer Science
- Oper. Res. Lett.
- 2001

A branch-and-cut algorithm for the Maximum Cardinality Stable Set problem is proposed and a computational experience on the DIMACS benchmark graphs validates the effectiveness of the approach. Expand

Finding a Maximum Independent Set

- Mathematics, Computer Science
- SIAM J. Comput.
- 1977

An algorithm is presented which finds a maximum independent set in an n-vertex graph in 0($2^{n/3}$) time and can thus handle graphs roughly three times as large as could be analyzed using a naive algorithm. Expand

Finding a Maximum Clique in an Arbitrary Graph

- Mathematics, Computer Science
- SIAM J. Comput.
- 1986

A new type of branch and bound procedure for finding a maximum clique in an arbitrary graph G, which discusses computational experience on randomly generated graphs with up to 400 vertices and 30,000 edges. Expand

A semidefinite programming-based heuristic for graph coloring

- Computer Science, Mathematics
- Discret. Appl. Math.
- 2008

A self-contained presentation of the role of the Lovasz @q-function in obtaining heuristics for the graph coloring problem is presented, which could be useful for coloring medium sized graphs as numerical results on DIMACS benchmark graphs indicate. Expand