## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Thursday, September 21, 2017, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Manuela Fischer

## Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching

We present a deterministic distributed algorithm that computes a $(2\Delta−1)$-edge-coloring, or even list-edge-coloring, in any n-node graph with maximum degree $\Delta$, in $O(\log^8\Delta \log n)$ rounds. This answers one of the long-standing open questions of distributed graph algorithms from the late 1980s, which asked for a polylogarithmic-time algorithm. See, e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and Elkin. The previous best round complexities were $2^{O(\sqrt{\log n})}$ by Panconesi and Srinivasan [STOC'92] and $\tilde{O}(\sqrt{\Delta})+O(log^*n)$ by Fraigniaud, Heinrich, and Kosowski [FOCS'16].
A corollary of our deterministic list-edge-coloring also improves the randomized complexity of $(2\Delta−1)$-edge-coloring to $poly(\log\log n)$ rounds.
The key technical ingredient is a deterministic distributed algorithm for hypergraph maximal matching, which we believe will be of interest beyond this result. In any hypergraph of rank r --- where each hyperedge has at most r vertices --- with n nodes and maximum degree $\Delta$, this algorithm computes a maximal matching in $O(r^5 \log^{6+\log r} \Delta \log n)$ rounds.

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