Moser-Tardos Algorithm: Beyond Shearer's Bound

Kun He, Qian Li, Xiaoming Sun

Abstract: In a seminal paper (Moser and Tardos, JACM'10), Moser and Tardos developed a simple and powerful algorithm to find solutions to constraint satisfaction problems. Kolipaka and Szegedy (Kolipaka and Szegedy, STOC'11) proved that the Moser-Tardos algorithm is efficient up to the tight condition of the abstract Lovász Local Lemma, known as Shearer's bound. A fundamental problem around the LLL is whether the efficient 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 efficient region of the Moser-Tardos algorithm indeed goes beyond the Shearer's bound of the underlying dependency graph, if the graph is not chordal. This “chordal condition” is sufficient and necessary, since it has been shown that Shearer's bound exactly characterizes the efficient region for chordal dependency graph (Kolipaka and Szegedy, STOC'11; He, Li, Liu, Wang and Xia, FOCS'17). Moreover, we demonstrate that the efficient region can exceed Shearer's bound by a constant amount by explicitly calculating the gaps on several infinite lattices.The core of our proof is a new criterion on the efficiency of the Moser-Tardos algorithm which takes the intersection between dependent events into consideration. Our criterion is strictly larger than Shearer's bound whenever there exist two dependent events with non-empty intersection. 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).* The full version of the paper can be accessed at https://arxiv.org/abs/2111.06527