The asymptotic analysis of different crossover and mutation operators helps in choosing the better evolutionary operator to minimize the problem search space and computational complexity. The present paper focused on the asymptotic analysis of some well-known and recent evolutionary operators for finding the chromatic number. The chromatic number of a graph G is defined as the minimum number of colors required to color the vertex set V(G) so that no two adjacent vertices are of the same color, and different approximations and evolutionary methods can find it. The graph coloring problem is an NP-hard combinatorial optimization problem and can be applied to various engineering applications. Our experiments show that our approaches operate efficiently on inputs too large to fit on a single GPU and scale up to graphs with 76.7 billion edges running on 128 GPUs. In addition, we propose a novel heuristic to reduce communication for recoloring in distributed graph coloring. We further extend our approaches to compute distance-2 and partial distance-2 colorings, giving the first known distributed, multi-GPU algorithm for these problems. The on-node parallel coloring uses implementations in KokkosKernels, which provide parallelization for both multicore CPUs and GPUs. and the shared-memory algorithms of Deveci et al. We present several MPI+GPU coloring approaches based on the distributed coloring algorithms of Gebremedhin et al. Many algorithms exist for graph coloring on a single GPU or in distributed memory, but to the best of our knowledge, hybrid MPI+GPU algorithms have been unexplored until this work. Graph coloring is often used in parallelizing scientific computations that run in distributed and multi-GPU environments it identifies sets of independent data that can be updated in parallel. This is the full version of a paper published at ACM/IEEE Supercomputing'20 under the same title Our degeneracy ordering relaxation is of separate interest for algorithms outside the context of coloring. In addition to provable guarantees, the developed algorithms have competitive run-times for several real-world graphs, while almost always providing superior coloring quality. For example, one of our algorithms ensures polylogarithmic depth and a bound on the number of used colors that is superior to all other parallelizable schemes, while maintaining work-efficiency. This simple idea enables significant benefits in several key aspects of graph coloring.
This introduces a tunable amount of parallelism into the degeneracy ordering that is otherwise hard to parallelize. The key idea is to design a relaxation of the vertex degeneracy order, a well-known graph theory concept, and to color vertices in the order dictated by this relaxation. We develop the first parallel graph coloring heuris-tics with strong theoretical guarantees on work and depth and coloring quality.
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