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A goal of this paper is to efficiently adapt the best ingredients of the graph colouring techniques to an NPhard satellite range scheduling problem, called MuRRSP. We propose two new heuristics for the MuRRSP, where as many jobs as possible have to be scheduled on several resources, while respecting time and capacity constraints. In the permutation solution space, which is widely used by other researchers, a solution is represented by a permutation of the jobs, and a schedule builder is needed to generate and evaluate a feasible schedule from the permutation. On the contrary, our heuristics are based on the solution space which contains all the feasible schedules. Based on the similarities between the graph colouring problem and the MuRRSP, we show that the latter solution space has significant advantages. A tabu search and an adaptive memory algorithms are designed to tackle the MuRRSP. These heuristics are derived from efficient graph colouring methods. Numerical experiments, performed on large, realistic, and challenging instances, showed that our heuristics are very competitive, robust, and outperform algorithms based on the permutation solution space

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Let G = (V, E) be a graph with vertex set V and edge set E. The k-coloring problem is to assign a color (a number chosen in {1,. .. , k}) to each vertex of G so that no edge has both endpoints with the same color. We present a new local search algorithm, called Variable Space Search (VSS), which we apply to the k-coloring problem. VSS extends the Formulation Space Search (FSS) methodology by considering several non equivalent formulations of a same problem, each one being associated with a set of neighborhoods and an objective function. The search moves from one formulation to another when it is blocked at a local optimum with a given formulation. The k-coloring problem is thus solved by combining different formulations of the problem which are not equivalent, in the sense that some constraints are possibly relaxed in one search space and always satisfied in another. We show that the proposed algorithm improves on every local search used independently (i.e., with a unique search space), and is competitive with the currently best coloring methods, which are complex hybrid evolutionary algorithms.

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