Sublinear Algorithms for (Delta + 1) Vertex Coloring

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Claim: sampling O(log n) random colors per vertex (a palette sparsification) suffices, with high probability, to retain a Delta+1 coloring. Tools: use the HSS decomposition to split sparse vs. almost-clique regions and build a conflict graph of size ~n log^2 n; information-theoretically this reduces edges from n·Delta to ~n log^2 n. Consequences: randomized algorithms give optimal O(n·sqrt n) time/queries, single-pass O~(n) streaming (even dynamic) space, and constant-round MPC (one round with public randomness); algorithmic step uses matching routines to make it efficient.

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" chapters (16)

1. 0:00Intro
2. 0:23Graph Coloring
3. 2:17The Greedy Algorithm for (A+1) Coloring
4. 4:09Motivating Question
5. 6:29Our Results Sublinear Time Algorithms
6. 7:58Our Results Streaming Algorithms
7. 10:55Our Results MPC Algorithms
8. 12:50Palette Sparsification: An Illustration
9. 16:28A Sight Reformulation
10. 21:31Handling Graphs that are Far From Cliques
11. 24:10General Proof?
12. 24:58A Network Decomposition
13. 27:43Proof Strategy of Palette Sparsification Theorem
14. 31:47Proof Strategy: An Illustration
15. 33:30The Sublinear Algorithms
16. 36:22Concluding Remarks