Abstract. Chromatic number, chromatic sum and chromatic sum number are important graph coloring characteristics. The paper proves that a parallel metaheuristic like the parallel genetic algorithm (PGA) can be efficiently used for computing approximate sum colorings and finding upper bounds for chromatic sums and chromatic sum numbers for hardto-color graphs. Suboptimal sum coloring with PGA gives usually much closer upper bounds then theoretical formulas known from the literature.

Abstract. In this paper a new parallel genetic algorithm for coloring graph vertices is presented. In the algorithm we apply a migration model of parallelism and de ne two new recombination operators: SPPX and CEX. For comparison two recently proposed crossover operators: UISX and GPX are selected. The performance of the algorithm is veri ed by computer experiments on a set of standard graph coloring instances.

Abstract.A hardware-oriented algorithm for generating permutations is presented that takes as a theoretic base an iterative decomposition of the symmetric group Sn into cosets. It generates permutations in a new order. Simple ranking and unranking algorithms are given. The construction of a permutation generator is proposed which contains a cellular permutation network as a main component. The application of the permutation generator for solving a class of combinatorial problems on parallel computers is suggested.

Abstract. The paper describes an application of Parallel Simulated Annealing (PSA) for solving one of the most studied NP-hard optimization problems: Graph Coloring Problem (GCP). Synchronous master-slave model with periodic solution update is being used. The paper contains description of the method, recommendations for optimal parameters settings and summary of results obtained during algorithm's evaluation. A comparison of our novel approach to a PGA metaheuristic proposed in the literature is given. Finally, directions for further work in the subject are suggested.