### Abstract

How to color 3 colorable graphs with few colors is a problem of longstanding interest. The best polynomial-time algorithm uses n ^{0.2072} colors. There are no indications that coloring using say O(log n) colors is hard. It has been suggested that SDP hierarchies could be used to design algorithms that use n^{ε} colors for arbitrarily small ε > 0. We explore this possibility in this paper and find some cause for optimism. While the case of general graphs is till open, we can analyse the Lasserre relaxation for two interesting families of graphs. For graphs with low threshold rank (a class of graphs identified in the recent paper of Arora, Barak and Steurer on the unique games problem), Lasserre relaxations can be used to find an independent set of size Ω(n) (i.e., progress towards a coloring with O(log n) colors) in n^{O(D)} time, where D is the threshold rank of the graph. This algorithm is inspired by recent work of Barak, Raghavendra, and Steurer on using Lasserre Hierarchy for unique games. The algorithm can also be used to show that known integrality gap instances for SDP relaxations like strict vector chromatic number cannot survive a few rounds of Lasserre lifting, which also seems reason for optimism. For distance transitive graphs of diameter Δ, we can show how to color them using O(log n) colors in n ^{2O(Δ)} time. This family is interesting because the family of graphs of diameter O(1/ε) is easily seen to be complete for coloring with n^{ε} colors. The distance-transitive property implies that the graph "looks" the same in all neighborhoods. The full version of this paper can be found at: http://www.cs.princeton.edu/~rongge/LasserreColoring.pdf.

Original language | English (US) |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization |

Subtitle of host publication | Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings |

Pages | 1-12 |

Number of pages | 12 |

DOIs | |

State | Published - Sep 8 2011 |

Event | 14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011 - Princeton, NJ, United States Duration: Aug 17 2011 → Aug 19 2011 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 6845 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011 |
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Country | United States |

City | Princeton, NJ |

Period | 8/17/11 → 8/19/11 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings*(pp. 1-12). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6845 LNCS). https://doi.org/10.1007/978-3-642-22935-0_1