# 3-Coloring in Time O(1.3289^n)

@article{Beigel20053ColoringIT, title={3-Coloring in Time O(1.3289^n)}, author={Richard Beigel and David Eppstein}, journal={J. Algorithms}, year={2005}, volume={54}, pages={168-204} }

We consider worst case time bounds for several NP-complete problems, based on a constraint satisfaction (CSP) formulation of these problems: (a, b)-CSP instances consist of a set of variables, each with up to a possible values, and constraints disallowing certain b-tuples of variable values; a problem is solved by assigning values to all variables satisfying all constraints, or by showing that no such assignment exist. 3-SAT is equivalent to (2, 3)-CSP while 3-coloring and various related…

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## 141 Citations

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## References

SHOWING 1-10 OF 50 REFERENCES

3-coloring in time 0(1.3446/sup n/): a no-MIS algorithm

- Mathematics, Computer ScienceProceedings of IEEE 36th Annual Foundations of Computer Science
- 1995

This work gives a fast algorithm for (3,2)-SSS and uses it to improve the time bounds for solving the other problems listed above.

Algorithms for k-colouring and finding maximal independent sets

- Mathematics, Computer ScienceSODA '03
- 2003

This extended abstract proves that the number of maximal independent sets of size at most k (k-MIS's) in a graph is at most d 1)dk-nd n-(d-1)k for any d > 4, and constructs algorithms for finding the chromatic number using exponential space, whereas the other algorithms use only polynomial space.

A Spectral Technique for Coloring Random 3-Colorable Graphs

- Computer Science, MathematicsSIAM J. Comput.
- 1997

A polynomial-time algorithm that finds a proper 3-coloring of G3n,p,3 with high probability, whenever p $\geq$ c/n, where c is a sufficiently large absolute constant.

A Spectral Technique for Coloring Random 3-Colorable Graphs

- 1994

18 until the number of vertices left is O(n=d), and color the remaining vertices in an arbitrary manner. 7. The existence of an approximation algorithm based on the spectral method for coloring…

A Note on the Complexity of the Chromatic Number Problem

- Mathematics, Computer ScienceInf. Process. Lett.
- 1976

It is shown, by very simple arguments, that the chromatic number of an arb?rary graph to within a given factor r < 2 has also been shown to “be Nkomplete” and there is very little prospect of fmding an efficient, i.e. polynomialbounded, algorithm for the general problem, although some special cases can be solved effkien?ly.

Finding maximum independent sets in sparse and general graphs

- Mathematics, Computer ScienceSODA '99
- 1999

The fastest MIS algorithms for sparse graphs and for graphs whose degree is bounded by 3 or by 4 are obtained, and the fastest MIS algorithm for bounded-degree graphs is obtained.

Small Maximal Independent Sets and Faster Exact Graph Coloring

- Computer Science, MathematicsJ. Graph Algorithms Appl.
- 2003

It is shown how to compute the exact chromatic number of a graph in time O(4/3 + 34/3/4)n, improving a previous O((1 + 31/3)n) ≅ 2.4422n algorithm of Lawler (1976).

A New Approach on Solving 3-Satisfiability

- Computer ScienceAISMC
- 1996

An algorithm for solving the 3-satisfiability problem is described and it is shown that the algorithm on subquadrangles guarantees a solution in time less than O(1.476n), which improves the current well-known 3-Satisfiability algorithms.

Two new upper bounds for SAT

- Mathematics, Computer ScienceSODA '98
- 1998

A new transformation rule is presented and two algorithms using this rule have the bounds 2°.30897K and 2.‘0537L respectively.

Systematic generation of very hard cases for graph 3-colorability

- Computer ScienceProceedings of 7th IEEE International Conference on Tools with Artificial Intelligence
- 1995

We present a simple generation procedure which turns out to be an effective source of very hard cases for graph 3-colorability. The graphs distributed according to this generation procedure are much…