Moser-Tardos Algorithm: Beyond Shearer's Bound

Abstract

A. In a seminal paper (Moser and Tardos, JACM'10), Moser and Tardos developed a simple and powerful algorithm to nd solutions to combinatorial problems in the variable Lovász Local Lemma (LLL) se ing. Kolipaka and Szegedy (Kolipaka and Szegedy, STOC'11) proved that the Moser-Tardos algorithm is e cient up to the tight condition of the abstract Lovász Local Lemma, known as Shearer's bound. A fundamental problem around LLL is whether the e cient region of the Moser-Tardos algorithm can be further extended.In this paper, we give a positive answer to this problem. We show that the e cient region of the Moser-Tardos algorithm goes beyond the Shearer's bound of the underlying dependency graph, if the graph is not chordal. Otherwise, the dependency graph is chordal, and it has been shown that Shearer's bound exactly characterizes the e cient region for such graphs (Kolipaka and Szegedy, STOC'11; He, Li, Liu, Wang and Xia, FOCS'17).Moreover, we demonstrate that the e cient region can exceed Shearer's bound by a constant by explicitly calculating the gaps on several in nite la ices.e core of our proof is a new criterion on the e ciency of the Moser-Tardos algorithm which takes the intersection between dependent events into consideration. Our criterion is strictly be er than Shearer's bound whenever the intersection exists between dependent events. Meanwhile, if any two dependent events are mutually exclusive, our criterion becomes the Shearer's bound, which is known to be tight in this situation for the Moser-Tardos algorithm (Kolipaka and Szegedy, STOC'11; Guo, Jerrum and Liu, JACM'19).

Comparison of Two Convergence Criteria for the Variable-Assignment Lopsided Lovasz Local Lemma

“…Namely, they found an explicit dependency graph and vector of probabilities where the Shearer criterion is violated yet any variable-assignment realization must have a satisfying assignment. Later work [11] provided a more systematic description of which dependency graphs were satisfiable in the variable-assignment setting.…”

“…. , m} and B and an edge on (i, B) when i ∈ var(B), and [11] derives conditions in terms of the probabilities that certain neighboring bad-events hold simultaneously.…”