We develop the first parallel graph coloring heuristics with strong theoretical guarantees on work and depth and coloring quality. 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. This introduces a tunable amount of parallelism into the degeneracy ordering that is otherwise hard to parallelize. This simple idea enables significant benefits in several key aspects of graph coloring. 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 workefficiency. In addition to provable guarantees, the developed algorithms have competitive run-times for several real-world graphs, while almost always providing superior coloring quality. Our degeneracy ordering relaxation is of separate interest for algorithms outside the context of coloring.

We present the first near-linear work and poly-logritharithmic depth algorithm for computing a minimum cut in a graph, while previous parallel algorithms with poly-logarithmic depth required at least quadratic work in the number of vertices.In a graph with n vertices and m edges, our algorithm computes the correct result with high probability in O(m log 4 n) work and O(log 3 n) depth. This result is obtained by parallelizing a data structure that aggregates weights along paths in a tree and by exploiting the connection between minimum cuts and approximate maximum packings of spanning trees.In addition, our algorithm improves upon bounds on the number of cache misses incurred to compute a minimum cut.

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