## GRAPH COLOURING PROBLEMS AND THEIR APPLICATIONS IN SCHEDULING (2003)

Citations: | 7 - 0 self |

### BibTeX

@MISC{Marx03graphcolouring,

author = {Dániel Marx},

title = {GRAPH COLOURING PROBLEMS AND THEIR APPLICATIONS IN SCHEDULING},

year = {2003}

}

### OpenURL

### Abstract

Graph colouring and its generalizations are useful tools in modelling a wide variety of scheduling and assignment problems. In this paper we review several variants of graph colouring, such as precolouring extension, list colouring, multicolouring, minimum sum colouring, and discuss their applications in scheduling.

### Citations

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(Show Context)
Citation Context ...heduling problem can be modelled as an edge colouring of this graph: we have to assign colours to the edges in such a way that every colour appears at most once at a vertex. Edge colouring is NP-hard =-=[3]-=-, but there are good approximation algorithms. The maximum degree of the graph is an obvious lower bound on the number of colours needed to colour the edges of the graph. On the other hand, if there a... |

63 | On chromatic sums and distributed resource allocation
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(Show Context)
Citation Context ...ees, and edges of trees, minimum sum colouring is NP-hard on most classes of graphs. On the other hand, it turns out that the sum of the colouring is easier to approximate than the makespan (see e.g. =-=[17, 18]-=- for approximation results). The reason for this is that the sum of the colouring and the makespan of the colouring behave very differently when a small part of the graph is recoloured. If we recolour... |

45 |
On the 1.1 edge-coloring of multigraphs
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(Show Context)
Citation Context ...there are no two tasks that require the same two processors), then Vizing’s Theorem gives an efficient method for obtaining a � 1 -edge colouring. If multiple edges are allowed, then the algorithm of =-=[4]-=- gives a 1�1-approximate solution. Frequency assignment. Assume that we have a number of radio stations, identified by x and y coordinates in the plane. We have to assign a frequency to each station, ... |

40 | Complexity of scheduling multi-processor tasks with prespecified processor allocations
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(Show Context)
Citation Context ...t work on two jobs at the same time. For example, such biprocessor tasks arise when we want to schedule file transfers between processors [1] or in the case of mutual diagnostic testing of processors =-=[2]-=-. Consider the graph whose vertices correspond to the processors, and if there is a task that has to be executed on processors i and j, then we add an edge between the two corresponding vertices. Now ... |

38 | S.Shende, Static Frequency Assignment in Cellular Networks, Algorithmica 29
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(Show Context)
Citation Context ...ndow. On the other hand, in preemptive multicolouring we assume that the jobs can be interrupted, hence the set of colours assigned to a vertex can be arbitrary, it does not have to be continuous. In =-=[6]-=- the frequency assignment problem discussed in Section 1 is generalized, we have to assign a predefined number of frequencies to each base station. The authors adopt the hexagon model, which means tha... |

36 | On coloring unit disk graphs
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(Show Context)
Citation Context ...onds to a disk in the plane with unit diameter, and two vertices are connected if and only if the corresponding disks intersect. A 3-approximation algorithm for colouring unit disk graphs is given in =-=[5]-=-, yielding a 3-approximation for the frequency assignment problem. 2. Multicolouring A natural generalization of the basic setup introduced in Section 1 is to consider jobs that require more than one ... |

35 | Precoloring extension. III. Classes of perfect graphs
- Hujter, Tuza
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(Show Context)
Citation Context ... a preassigned colour, and we have to solve the precolouring extension problem: extended the colouring of these vertices to the whole graph, using the minimum number of colours. BIRÓ, HUJTER and TUZA =-=[7, 8, 9]-=- started a systematic study of precolouring extension. In [7], the aircraft scheduling problem discussed in Section 1 is extended. There is a maintenance period for each aircraft, during which it cann... |

33 |
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(Show Context)
Citation Context ...theory is to minimize the sum of completion times of the jobs, which is the same as minimizing the average completion time. The corresponding colouring problem is minimum sum colouring, introduced in =-=[16]-=-: we are looking for a colouring of the conflict graph such that the sum of the colours assigned to the vertices is minimal. Apart from trees, partial k-trees, and edges of trees, minimum sum colourin... |

27 | Sum multicoloring of graphs
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(Show Context)
Citation Context ... the vertices. It is clear that the sum of the finish times in a multicolouring is equal to the sum of completion times in the corresponding schedule. This variant of multicolouring was introduced in =-=[19]-=-, where approximation algorithms are given for various classes of graphs. The preemptive and non-preemptive versions of the problem can have very different complexity: while the non-preemptive version... |

23 | Generalized coloring for tree-like graphs
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(Show Context)
Citation Context ... in certain time slots, or if it can be processed only by certain machines. Using standard dynamic programming techniques, list colouring can be solved in polynomial time on trees and partial k-trees =-=[11]-=-. By combining dynamic programming with a clever use matching, list colouring can be solved on the edges of trees as well [12]. The multicolouring concept introduced in Section 2 can be applied for li... |

23 | Tools for multicoloring with applications to planar graphs and partial k-trees
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(Show Context)
Citation Context ...ity: while the non-preemptive version can be solved in polynomial time for trees [20], the preemptive version is NP-hard for binary trees [13], but has a polynomial time approximation scheme [20]. In =-=[21]-=- polynomial time approximation schemes are given for partial k-trees and planar graphs as well. Unlike minimum sum colouring, the multicolouring version of the problem is NP-hard on the edges of trees... |

15 |
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(Show Context)
Citation Context ... a preassigned colour, and we have to solve the precolouring extension problem: extended the colouring of these vertices to the whole graph, using the minimum number of colours. BIRÓ, HUJTER and TUZA =-=[7, 8, 9]-=- started a systematic study of precolouring extension. In [7], the aircraft scheduling problem discussed in Section 1 is extended. There is a maintenance period for each aircraft, during which it cann... |

11 |
Edge-chromatic sum of trees and bounded cyclicity graphs
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(Show Context)
Citation Context ...t colouring can be solved in polynomial time on trees and partial k-trees [11]. By combining dynamic programming with a clever use matching, list colouring can be solved on the edges of trees as well =-=[12]-=-. The multicolouring concept introduced in Section 2 can be applied for list colourings as well: each vertex has an integer demand x Ú , and vertex Ú has to receive a set of x Ú colours from its list ... |

11 | The complexity of tree multicolorings
- Marx
- 2002
(Show Context)
Citation Context ... Ú colours from its list of colours. The algorithm for list colouring trees and partial k-trees does not generalize for the multicolouring case, as the problem is NP-complete already for binary trees =-=[13]-=-. On the other hand, list edge multicolouring can be solved in polynomial time on trees: using standard techniques, the good characterization theorem of MARCOTTE and SEYMOUR [14] can be turned into a ... |

10 | Z.: Precoloring extension. II. Graph classes related to bipartite graphs
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(Show Context)
Citation Context ... a preassigned colour, and we have to solve the precolouring extension problem: extended the colouring of these vertices to the whole graph, using the minimum number of colours. BIRÓ, HUJTER and TUZA =-=[7, 8, 9]-=- started a systematic study of precolouring extension. In [7], the aircraft scheduling problem discussed in Section 1 is extended. There is a maintenance period for each aircraft, during which it cann... |

10 |
Extending an edge-coloring
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(Show Context)
Citation Context ...eady for binary trees [13]. On the other hand, list edge multicolouring can be solved in polynomial time on trees: using standard techniques, the good characterization theorem of MARCOTTE and SEYMOUR =-=[14]-=- can be turned into a polynomial time algorithm. This result is generalized in [15] to a slightly more general class of graphs, that includes odd cycles. Moreover, a randomized algorithm is given for ... |

8 | Precoloring extension on chordal graphs
- Marx
- 2004
(Show Context)
Citation Context ...RX solved in polynomial time if every colour is used only once in the precolouring, that is, if every aircraft has only one maintenance interval (the latter result is generalized to chordal graphs in =-=[10]-=-). 4. List Colouring In the list colouring problem each vertex Ú has a list of available colours, and we have to find a colouring where the colour of each vertex is taken from its list of available co... |

3 | List edge multicoloring in graphs with few cycles
- Marx
(Show Context)
Citation Context ...ed in polynomial time on trees: using standard techniques, the good characterization theorem of MARCOTTE and SEYMOUR [14] can be turned into a polynomial time algorithm. This result is generalized in =-=[15]-=- to a slightly more general class of graphs, that includes odd cycles. Moreover, a randomized algorithm is given for an even more general class of graphs, including even cycles. 5. Minimum Sum Colouri... |

1 |
Scheduling File Transfers
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(Show Context)
Citation Context ...n two preassigned processors simultaneously. A processor cannot work on two jobs at the same time. For example, such biprocessor tasks arise when we want to schedule file transfers between processors =-=[1]-=- or in the case of mutual diagnostic testing of processors [2]. Consider the graph whose vertices correspond to the processors, and if there is a task that has to be executed on processors i and j, th... |

1 | Minimum Sum Multicoloring on the Edges of Trees, 1st Workshop on Approximation and Online Algorithms (WAOA - MARX - 2003 |