## S1-47–1 Efficiency issues in the RLF heuristic for graph coloring

Citations: | 2 - 0 self |

### BibTeX

@MISC{Chiar_s1-47–1efficiency,

author = {Marco Chiar and Giulia Galbiati and Stefano Gual},

title = {S1-47–1 Efficiency issues in the RLF heuristic for graph coloring},

year = {}

}

### OpenURL

### Abstract

This paper presents an efficient implementation of the well-known Recursive Largest First (RLF) algorithm of Leighton to find heuristic solutions to the classical graph coloring problem. The main features under study are a lazy computation of the induced vertex degree and the use of efficient data structures. Computational experiments show that the lazy feature leads to a novel implementation that is faster than previous implementations on graphs with high density. Cache-misses are instead determinant for the assessment of data structures. 1

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Citation Context ...5 Udine, Italy, July 25–28, 2011S1-47–6 MIC 2011: The IX Metaheuristics International Conference Uniform Random Graphs: These graphs are generated according the G(n, p) Erdös-Rényi model (see, e.g., =-=[7]-=-). Weight Biased Graphs: These graphs were proposed in [6] where they were shown to be among the hardest to color. They are designed to avoid large cliques and at the same time to favor flatness crite... |

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Citation Context ...oloring exists. In the optimization version, we are asked for the smallest number k, called the chromatic number χ(G), for which a k-coloring exists. In general, the k-coloring problem is NP-complete =-=[9]-=-. For the optimization version, bad results exist also in terms of approximability, for example, a ratios of n1− cannot be achieved in polynomial time unless ZPP=NP [8]. A number of polynomial time co... |

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Citation Context ...coloring problem is NP-complete [9]. For the optimization version, bad results exist also in terms of approximability, for example, a ratios of n1− cannot be achieved in polynomial time unless ZPP=NP =-=[8]-=-. A number of polynomial time constructive algorithms to solve heuristically the graph coloring problem have been proposed in the literature. These algorithms do not exhibit guaranteed approximation r... |

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Citation Context ...duced by the colored vertices (ties are broken by the original degree, in non increasing order). A third constructive algorithm is the so-called Recursive Largest First (RLF) algorithm, introduced in =-=[10]-=-. This algorithm has a different strategy: it sequentially colors stable sets, that is, it sequentially builds sets of vertices that can take the same color. Computational studies on these algorithms ... |

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Citation Context ...Metaheuristics International Conference Uniform Random Graphs: These graphs are generated according the G(n, p) Erdös-Rényi model (see, e.g., [7]). Weight Biased Graphs: These graphs were proposed in =-=[6]-=- where they were shown to be among the hardest to color. They are designed to avoid large cliques and at the same time to favor flatness criteria such as small variations in vertex degree and hidden c... |

16 |
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Citation Context ... [1], instead of a standard list implementation (e.g., the class list of the Standard Template Library in C++). Cache-misses are reduced because the arrays are allocated in contiguous areas of memory =-=[3]-=-. Udine, Italy, July 25–28, 2011 464MIC 2011: The IX Metaheuristics International Conference S1-47–5 Algorithm 4 - function LAZYSELECTVERTEX(G, P, U) In G =(V,E) : input graph Var P : set of potentia... |

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Citation Context ...t of the neighbor j. These two pointers allow for constant time edge removals. In order to minimize the chances of cache-misses we use an array-based implementation of linked lists, as recommended in =-=[1]-=-, instead of a standard list implementation (e.g., the class list of the Standard Template Library in C++). Cache-misses are reduced because the arrays are allocated in contiguous areas of memory [3].... |

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4 |
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(Show Context)
Citation Context ...ly builds sets of vertices that can take the same color. Computational studies on these algorithms show that RLF clearly outperforms the other two in terms of quality on a wide range of graph classes =-=[5]-=-. However, RLF comes with a higher computational cost, having a O(n3 ) worst-case complexity, instead of O(n2 ) of the other two heuristics. In spite of its best results in terms of quality, the RLF h... |

1 |
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(Show Context)
Citation Context ... terms of quality, the RLF heuristic has been seldom used in the literature on metaheuristics methods for the GCP, both as a reference method and as a building block for metaheuristic algorithms (see =-=[4]-=- for a list of references). In this paper we study efficient implementations of this algorithm. We put forward LAZY RLF, a variant of RLF that exploits possible savings in the computations during the ... |