## New approximation algorithms for graph coloring (1994)

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Venue: | J. ACM |

Citations: | 37 - 2 self |

### BibTeX

@ARTICLE{Blum94newapproximation,

author = {Avrim Blum},

title = {New approximation algorithms for graph coloring},

journal = {J. ACM},

year = {1994}

}

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### Abstract

The problem of coloring a graph with the minimum number of colors is well known to be NP-hard, even restricted to k-colorable graphs for constant k ≥ 3. This paper explores the approximation problem of coloring k-colorable graphs with as few additional colors as possible in polynomial time, with special focus on the case of k = 3. The previous best upper bound on the number of colors needed for coloring 3-colorable n-vertex graphs in polynomial time was O ( √ n / √ log n) colors by Berger and Rompel, improving a bound of O ( √ n) colors by Wigderson. This paper presents an algorithm to color any 3-colorable graph with O(n 3/8 polylog(n)) colors, thus breaking an “O(n 1/2−o(1) ) barrier”. The algorithm given here is based on examining second-order neighborhoods of vertices, rather than just immediate neighborhoods of vertices as in previous approaches. We extend our results to improve the worst-case bounds for coloring k-colorable graphs for constant k> 3 as well. 1

### Citations

707 | Proof verification and hardness of approximation problems - Arora, Lund, et al. - 1998 |

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Graphs and Hypergraphs
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Citation Context ... the smallest k such that the graph can be k-colored. Graph coloring problems model a collection of scheduling problems such as examination scheduling and register allocation [Cha82][CAC + 81][BCKT89]=-=[Ber73]-=-. Graph coloring is also closely related to other combinatorial problems such as finding the maximum independent set in a graph (the largest set of vertices such that no two have an edge between them)... |

416 |
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Citation Context ...atic number of a graph is the smallest k such that the graph can be k-colored. Graph coloring problems model a collection of scheduling problems such as examination scheduling and register allocation =-=[Cha82]-=-[CAC + 81][BCKT89][Ber73]. Graph coloring is also closely related to other combinatorial problems such as finding the maximum independent set in a graph (the largest set of vertices such that no two h... |

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Citation Context ...dge removed from G has an endpoint in R is less than p. Since we remove N edges to create GN and each time we remove an edge the probability it has an endpoint in R is less than p, by Chernoff bounds =-=[AV79]-=- the probability we remove more than pN(1 + ffi) vertices from R is at most e \Gammaffi 2 \Omega\Gamma pN) . Since pN = \Omega\Gamma m) and we assume m AE 1=ffi 2 in the statement of the theorem, the ... |

154 |
A local-ratio theorem for approximating the weighted vertex cover problem
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Citation Context ...Sj, and therefore a vertex cover of size 1 2 jSj, we can algorithmically find an independent set of size ~\Omega (n 2=3 ) by applying a vertex-cover approximation algorithm due to Bar-Yehuda and Even =-=[BYE85]-=- and, independently, to Monien and Speckenmeyer [MS85]. (Their algorithms differ slightly but the bounds are essentially the same; a version of their algorithm is described in the appendix for complet... |

135 | Approximating maximum independent sets by excluding subgraphs
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Citation Context ...a performance guarantee---that is, a ratio of the number of colors used to the chromatic number--- of O(n(log log n) 2 =(log n) 3 ). This result is based upon an algorithm by Boppana and Halld'orsson =-=[BH90]-=- for the Independent Set problem which finds an independent set within an n=(log n) 2 factor of the maximum. The difficulty in improving the algorithmic results has motivated work on lower bounds for ... |

133 |
Coloring heuristics for register allocation
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Citation Context ...graph is the smallest k such that the graph can be k-colored. Graph coloring problems model a collection of scheduling problems such as examination scheduling and register allocation [Cha82][CAC + 81]=-=[BCKT89]-=-[Ber73]. Graph coloring is also closely related to other combinatorial problems such as finding the maximum independent set in a graph (the largest set of vertices such that no two have an edge betwee... |

73 | A still better performance guarantee for approximate graph coloring
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- 1993
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Citation Context ...ed it was clear that, say, O( p n= log 2 n) colors would be completely out of reach. For general graphs of arbitrary chromatic number, the best algorithmic result known to date is due to Halld'orsson =-=[Hal90]-=-. Halld'orsson's algorithm has a performance guarantee---that is, a ratio of the number of colors used to the chromatic number--- of O(n(log log n) 2 =(log n) 3 ). This result is based upon an algorit... |

68 |
Improving the performance guarantee for approximate graph coloring
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- 1983
(Show Context)
Citation Context ...om" k-chromatic graphs can be optimally colored with high probability. 1.1 Past work For graphs of constant chromatic number, the first nontrivial worst-case approximation result was due to Wigde=-=rson [Wig83]-=-. Wigderson gives an algorithm to color any n-vertex 3-colorable graph with O( p n) colors, and more generally to color any k-colorable graph with O(n 1\Gamma 1 k\Gamma1 ) colors. More recently, sever... |

66 | Almost all k-colorable graphs are easy to color
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- 1988
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Citation Context ...we do not consider here is to provide algorithms that find optimal colorings for large or nicely characterized subsets of the inputs. Work along this direction has been done by Kucera [Kuc77], Turner =-=[Tur88], Dyer and-=- Frieze [DF89], and Blum [Blu91]; in particular, these results show that large classes of random or "semi-random" k-chromatic graphs can be optimally colored with high probability. 1.1 Past ... |

58 |
Ramsey numbers and an approximation algorithm for the vertex cover problem
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Citation Context ...n algorithmically find an independent set of size ~\Omega (n 2=3 ) by applying a vertex-cover approximation algorithm due to Bar-Yehuda and Even [BYE85] and, independently, to Monien and Speckenmeyer =-=[MS85]-=-. (Their algorithms differ slightly but the bounds are essentially the same; a version of their algorithm is described in the appendix for completeness.) Their algorithm finds a vertex cover of size a... |

29 |
The solution of some random NP-hard problems in polynomial expected time
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- 1989
(Show Context)
Citation Context ...s to provide algorithms that find optimal colorings for large or nicely characterized subsets of the inputs. Work along this direction has been done by Kucera [Kuc77], Turner [Tur88], Dyer and Frieze =-=[DF89], and Blum-=- [Blu91]; in particular, these results show that large classes of random or "semi-random" k-chromatic graphs can be optimally colored with high probability. 1.1 Past work For graphs of const... |

25 |
A better performance guarantee for approximate graph coloring
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- 1990
(Show Context)
Citation Context ...color any n-vertex 3-colorable graph with O( p n) colors, and more generally to color any k-colorable graph with O(n 1\Gamma 1 k\Gamma1 ) colors. More recently, several researchers: Berger and Rompel =-=[BR88]-=-, Linial, Saks, and Wigderson [LSW], and Raghavan [Rag] independently improved this bound to O((n= log n) 1\Gamma 1 k\Gamma1 ) colors, which for k = 3 results in a coloring of 3-colorable graphs with ... |

21 | Algorithms for approximate graph coloring
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- 1992
(Show Context)
Citation Context ...rithms that find optimal colorings for large or nicely characterized subsets of the inputs. Work along this direction has been done by Kucera [Kuc77], Turner [Tur88], Dyer and Frieze [DF89], and Blum =-=[Blu91]; in parti-=-cular, these results show that large classes of random or "semi-random" k-chromatic graphs can be optimally colored with high probability. 1.1 Past work For graphs of constant chromatic numb... |

19 | Some tools for approximate 3-coloring
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- 1990
(Show Context)
Citation Context ...e in fact chosen randomly, and this motivation and the general flavor of the algorithms are given in Section 3. Some of the work in this paper has previously appeared in extended abstract form [Blu89]=-=[Blu90]-=-, and additional results with more detailed discussion appears in [Blu91]. 2 Notation, definitions, and previous algorithms In this section we review some standard graph-theoretic definitions and intr... |

15 |
An Õ(n0.4 )-approximation algorithm for 3-coloring (and improved approximation algorithms for k-coloring
- Blum
- 1989
(Show Context)
Citation Context ...aph were in fact chosen randomly, and this motivation and the general flavor of the algorithms are given in Section 3. Some of the work in this paper has previously appeared in extended abstract form =-=[Blu89]-=-[Blu90], and additional results with more detailed discussion appears in [Blu91]. 2 Notation, definitions, and previous algorithms In this section we review some standard graph-theoretic definitions a... |

9 |
Private communication
- Linial
- 1993
(Show Context)
Citation Context ... with O( p n) colors, and more generally to color any k-colorable graph with O(n 1\Gamma 1 k\Gamma1 ) colors. More recently, several researchers: Berger and Rompel [BR88], Linial, Saks, and Wigderson =-=[LSW]-=-, and Raghavan [Rag] independently improved this bound to O((n= log n) 1\Gamma 1 k\Gamma1 ) colors, which for k = 3 results in a coloring of 3-colorable graphs with O( p n= p log n) colors. The result... |

6 |
Expected behavior of graph colouring algorithms
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- 1977
(Show Context)
Citation Context ...tion issue that we do not consider here is to provide algorithms that find optimal colorings for large or nicely characterized subsets of the inputs. Work along this direction has been done by Kucera =-=[Kuc77], Turner [-=-Tur88], Dyer and Frieze [DF89], and Blum [Blu91]; in particular, these results show that large classes of random or "semi-random" k-chromatic graphs can be optimally colored with high probab... |

6 | allocation via coloring - Register - 1981 |

2 |
Randomized online graph coloring (preliminary version
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- 1990
(Show Context)
Citation Context ...ented in an on-line manner: graphs presented one vertex at a time in some arbitrary order, with the requirement that an algorithm color the vertex presented before the next one is shown. Vishwanathan =-=[Vis90]-=- presents an algorithm for such a model that uses a number of colors within a logarithmic factor of the Wigderson bound. 1.2 New results In this paper, we present an algorithm that uses a quite differ... |

2 | Editors) Graph Colourings. Longman Scientific - Nelson, Wilson - 1998 |

1 |
On the hardness of approximating minimization problems. Unpublished manuscript
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- 1992
(Show Context)
Citation Context ...ndependent set within an n=(log n) 2 factor of the maximum. The difficulty in improving the algorithmic results has motivated work on lower bounds for this problem. Very recently, Lund and Yannakakis =-=[LY92]-=-, based on work of Arora, Lund, Motwani, Sudan, and Szegedy [ALM + 92], have shown that for some ffl ? 0, the chromatic number cannot in general be approximated to a ratio better than n ffl unless P=N... |

1 | will describe the algorithm as an Independent Set approximation algorithm for the special case where the input n-vertex graph contains an independent set of at least 2 (1 \Gamma log n ) of its vertices. The output of the procedure is an independent set of - We |

1 | BH90] [Blu89] [Blu90] [Blu91] [BR88] [BYE85 - Graphs, North-Holland - 1973 |