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12 - Chromatic scheduling

Published online by Cambridge University Press:  05 May 2015

By
Dominique de Werra  and
Alain Hertz
Edited by
Lowell W. Beineke  and
Robin J. Wilson
Dominique de Werra
Affiliation:
Ecole Polytechnique Fédérale de Lausanne in Switzerland
Alain Hertz
Affiliation:
École Polytechnique Fédérale de Lausanne in Switzerland
Lowell W. Beineke
Affiliation:
Purdue University, Indiana
Robin J. Wilson
Affiliation:
The Open University, Milton Keynes
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Summary

Variations and extensions of the basic vertex-colouring and edge-colouring models have been developed to deal with increasingly complex scheduling problems. We present and illustrate them in specific situations where additional requirements are imposed. We include list-colouring, mixed graph colouring, co-colouring, colouring with preferences and bandwidth colouring, and we present applications of edge-colourings to open shop, school timetabling and sports scheduling problems. We also discuss balancing and compactness constraints which often appear in practical situations.

Introduction

We show here how graph colouring models may provide a natural tool for dealing with a variety of scheduling problems. Starting from the basic vertex-colouring model, we will introduce some variations and extensions that are motivated by their applications to some scheduling issues. In each case we give references for further results and for extensions of the various models presented. For algorithms, see Chapter 13.

In chromatic scheduling problems we have a collection V of items, such as operations of jobs to be performed. In V there are some pairs v, w that are subject to an incompatibility condition and we call E the set of such incompatibility pairs. These data are represented by the graph G = (V, E) in which the items are associated with the vertices and the incompatible pairs v,w with the edges vw between the corresponding vertices.

We also have a set C = {1, 2, …, k} of time periods (of unit duration). Assuming that each item (considered as an operation) has unit completion time, we may ask whether we can find a schedule taking the incompatibilities into account and using at most k periods of time. This is precisely the vertex k-colouring problem: there exists a feasible schedule if and only if the set V of vertices can be partitioned into subsets S1, S2, …, Sk, where each Si contains no two incompatible items.

In some instances, we may try to find the smallest set C of periods (that is, the smallest k) for which a schedule in time k = |C| exists.

Type
Chapter
Information
Topics in Chromatic Graph Theory , pp. 255 - 276
Publisher: Cambridge University Press
Print publication year: 2015

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