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We present a poly $log log n$ time randomized CONGEST algorithm for a natural class of Lovasz Local Lemma (LLL) instances on constant degree graphs. This implies, among other things, that there are no LCL problems with randomized complexity between $log n$ and poly $log log n$. Furthermore, we provide extensions to the network decomposition algorithms given in the recent breakthrough by Rozhon and Ghaffari [STOC2020] and the follow up by Ghaffari, Grunau, and Rozhon [SODA2021]. In particular, we show how to obtain a large distance separated weak network decomposition with a negligible dependency on the range of unique identifiers.
We show that any randomised Monte Carlo distributed algorithm for the Lovasz local lemma requires $Omega(log log n)$ communication rounds, assuming that it finds a correct assignment with high probability. Our result holds even in the special case of
We consider the task of designing Local Computation Algorithms (LCA) for applications of the Lov{a}sz Local Lemma (LLL). LCA is a class of sublinear algorithms proposed by Rubinfeld et al.~cite{Ronitt} that have received a lot of attention in recent
Let $Phi = (V, mathcal{C})$ be a constraint satisfaction problem on variables $v_1,dots, v_n$ such that each constraint depends on at most $k$ variables and such that each variable assumes values in an alphabet of size at most $[q]$. Suppose that eac
We develop tools for analyzing focused stochastic local search algorithms. These are algorithms which search a state space probabilistically by repeatedly selecting a constraint that is violated in the current state and moving to a random nearby stat
We study the problem of sampling an approximately uniformly random satisfying assignment for atomic constraint satisfaction problems i.e. where each constraint is violated by only one assignment to its variables. Let $p$ denote the maximum probabilit