Institute of Information Science, Academia Sinica

The Lovasz Local Lemma (LLL), introduced by Erdos and Lovasz in 1975, is a powerful tool of the probabilistic method that allows one to prove that a set of n "bad" events do not happen with non-zero probability, provided that the events have limited dependence. However, the LLL itself does not suggest how to find a point avoiding all bad events. Since the work of Beck (1991) there has been a sustained effort in finding a constructive proof (i.e. an algorithm) for the LLL or weaker versions of it. In a major breakthrough Moser and Tardos (2010) showed that a point avoiding all bad events can be found effciently. They also proposed a distributed/parallel version of their algorithm that requires O(log^2 n) rounds of communication in a distributed network.

In this talk I will present new distributed algorithms for the LLL that improve on both the effciency and simplicity of the Moser-Tardos algorithm. Let p bound the probability of any bad event and d be the maximum degree in the dependency graph of the bad events. 

1. When epd^2 < 1 we give a truly simple LLL algorithm running in O(log_{1/epd^2} n) rounds. 

2. Under the tighter condition ep(d + 1) < 1, we give a slightly slower algorithm running  in O(log^2d log_{1/ep(d+1)} n) rounds. 

3. Under the stronger condition that e^2 p (d+1)^4 2^d < 1, we give a sublogarithmic  algorithm running in O(log n / log log n) rounds. 

Although the conditions of the LLL are locally verifiable, we prove that any distributed  LLL algorithm requires [@BackSlash]Omega(log* n) rounds.

In many graph coloring problems the existence of a valid coloring is established by one or more applications of the LLL. Using our LLL algorithms, frugal coloring, defective coloring, coloring girth-4 (triangle-free) and girth-5 graphs, edge coloring, and list coloring can be obtained in logarithmic distributed rounds.

Joint work with Kai-Min Chung and Seth Pettie.