## 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: Tuesday, June 06, 2017, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Mohsen Ghaffari

## (Blackboard Talk) Improved LOCAL Algorithms for the Lovasz Local Lemma, with Ramifications on LOCAL Complexities

The Lovasz Local Lemma and its algorithmic aspects are certainly not unfamiliar to this crowd. How about LOCAL algorithms for the Lovasz Local Lemma? To my surprise, this topic entails a remarkable significance well beyond a wordplay.

In a very recent revelation, Chang and Pettie [arXiv 1704.06297] showed that any "local" problem that can be solved (randomly) in the LOCAL model in o(log n) rounds on bounded degree graphs can be solved in T_LLL(n) rounds, which denotes the LOCAL round complexity for solving (a relaxed version of) LLL on bounded degree graphs. There was only one problem: the best known bound remained T_LLL(n)=O(log n). But they had a dream! They conjectured that T_LLL(n)=O(log log n).

Making the first step towards their conjecture, and improving significantly on the O(log n)-round LLL algorithm of Chung, Pettie, and Su [2014], we prove that T_LLL(n)=2^{O(sqrt{log log n})}. Hence, any o(log n)-round random LOCAL algorithm for any "local'' problem on bounded degree graphs can be automatically sped up to run in 2^{O(sqrt{log log n})} rounds, that is, e.g., well below O(log^{0.001} n).

This is based on a joint work with Manuela Fischer.

Information for students and suggested topics for student talks