Deterministic Algorithms for the Lovász Local Lemma

Abstract

The Lovász Local Lemma [5] (LLL) is a powerful result in probability theory that states that the probability that none of a set of bad events happens is nonzero if the probability of each event is small compared to the number of events that depend on it. It is often used in combination with the probabilistic method for non-constructive existence proofs. A prominent application is to k-CNF formulas, where LLL implies that, if every clause in the formula shares variables with at most d ≤ 2 k /e other clauses then such a formula has a satisfying assignment. Recently, a randomized algorithm to efficiently construct a satisfying assignment was given by Moser [12]. Subsequently Moser and Tardos [13] gave a randomized algorithm to construct the structures guaranteed by the LLL in a very general algorithmic framework. We address the main problem left open by Moser and Tardos of derandomizing these algorithms efficiently. Specifically, for a k-CNF formula with m clauses and d ≤ 2 k/(1+ǫ) /e for some ǫ ∈ (0, 1), we give an algorithm that finds a satisfying assignment in timeÕ(m 2(1+1/ǫ) ). This improves upon the deterministic algorithms of Moser and of Moser-Tardos with running time m Ω(k 2 ) which is superpolynomial for k = ω(1) and upon other previous algorithms which work only for d ≤ 2 k/16 /e. Our algorithm works efficiently for a general version of LLL under the algorithmic framework of Moser and Tardos [13], and is also parallelizable, i.e., has polylogarithmic running time using polynomially many processors.

Focused Stochastic Local Search and the Lovász Local Lemma

“…To bound ρ f (σ, σ ′ ) in (22) we recall that RECOLOR assigns to each uncolored vertex u ∈ N v a random color from R v u (σ) and invoke Lemma 5.2 to derive the first equality in (23). For the second equality we observe that for every u ∈ N v , the set R v u is determined by the colors of the vertices in V \ N v .…”

Synchronization strings: explicit constructions, local decoding, and applications

“…We eventually want to show that our LLL argument satisfies the conditions required for polynomialtime deterministic algorithmic LLL specified in [7]. Namely, it suffices to certify two other properties in addition to (1).…”

“…Unless, l > 2 log c n that contradicts the minimality of our choice of l. Proof. To proof this, we will make use of the Lovász local lemma and deterministic algorithms proposed for it in [7]. We generate a random string R over an alphabet of size |Σ| = O(ε −2 ) and define bad event B i 1 ,l 1 ,i 2 ,l 2 as the event of intervals [i 1 , i 1 + l 1 ) and [i 2 , i 2 + l 2 ) violating the O(1/ε)long-distance synchronization string property over intervals of total length 2/ε 2 or more.…”

Synchronization Strings: Highly Efficient Deterministic Constructions over Small Alphabets

“…Proof. To prove this, we will use the Lovśz Local Lemma and its deterministic algorithm in [6]. Suppose the alphabet is Σ with |Σ| = q = c 1 ε −2 where c 1 is a constant.…”