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A goal of this paper is to efficiently adapt the best ingredients of the graph colouring techniques to an NP-hard 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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polynomial-time'' means
efficient''. That algorithm is sequential and deterministic. We have also known since the 1980s that the matching problem has efficient parallel algorithms if the use of randomness is allowed. Formally, it is in the class RNC, i.e., it has randomized algorithms that use polynomially many processors and run in polylogarithmic time. However, we do not know if randomness is necessary - that is, whether the matching problem is in the class NC.
In this thesis we show that the matching problem is in quasi-NC. That is, we give a deterministic parallel algorithm that runs in O(log^3 n) time on n^{O(log^2 n)} processors. The result is obtained by a derandomization of the Isolation Lemma for perfect matchings, which was introduced in the classic paper by Mulmuley, Vazirani and Vazirani to obtain an RNC algorithm. Our proof extends the framework of Fenner, Gurjar and Thierauf, who proved the analogous result in the special case of bipartite graphs. Compared to that setting, several new ingredients are needed due to the significantly more complex structure of perfect matchings in general graphs. In particular, our proof heavily relies on the laminar structure of the faces of the perfect matching polytope.