We provide a tight analysis that settles the round complexity of the well-studied parallel randomized greedy MIS algorithm, thus answering the main open question of Blelloch, Fineman, and Shun [SPAA’12]. The parallel/distributed randomized greedy Maximal Independent Set (MIS) algorithm works as follows. An order of the vertices is chosen uniformly at random. Then, in each round, all vertices that appear before their neighbors in the order are added to the independent set and removed from the graph along with their neighbors. The main question of interest is the number of rounds it takes until the graph is empty. This algorithm has been studied since 1987, initiated by Coppersmith, Raghavan, and Tompa [FOCS’87], and the previously best known bounds were O (log n ) rounds in expectation for Erdős-Rényi random graphs by Calkin and Frieze [Random Struc. Alg.’90] and O (log 2 n ) rounds with high probability for general graphs by Blelloch, Fineman, and Shun [SPAA’12]. We prove a high probability upper bound of O (log n ) on the round complexity of this algorithm in general graphs and that this bound is tight. This also shows that parallel randomized greedy MIS is as fast as the celebrated algorithm of Luby [STOC’85, JALG’86].

We provide a tight analysis which settles the round complexity of the well-studied parallel randomized greedy MIS algorithm, thus answering the main open question of Blelloch, Fineman, and Shun [SPAA’12]. The parallel/distributed randomized greedy Maximal Independent Set (MIS) algorithm works as follows. An order of the vertices is chosen uniformly at random. Then, in each round, all vertices that appear before their neighbors in the order are added to the independent set and removed from the graph along with their neighbors. The main question of interest is the number of rounds it takes until the graph is empty. This algorithm has been studied since 1987, initiated by Coppersmith, Raghavan, and Tompa [FOCS’87], and the previously best known bounds were O(log n) rounds in expectation for Erdős-Rényi random graphs by Calkin and Frieze [Random Struc. & Alg. ’90] and O(log2 n) rounds with high probability for general graphs by Blelloch, Fineman, and Shun [SPAA’12]. We prove a high probability upper bound of O(log n) on the round complexity of this algorithm in general graphs, and that this bound is tight. This also shows that parallel randomized greedy MIS is as fast as the celebrated algorithm of Luby [STOC’85, JALG’86].

In his seminal paper from 1952 Dirac showed that the complete graph on n ≥ 3 vertices remains Hamiltonian even if we allow an adversary to remove ⌊n/2⌋ edges touching each vertex. In 1960 Ghouila-Houri obtained an analogue statement for digraphs by showing that every directed graph on n ≥ 3 vertices with minimum in- and out-degree at least n/2 contains a directed Hamilton cycle. Both statements quantify the robustness of complete graphs (digraphs) with respect to the property of containing a Hamilton cycle. A natural way to generalize such results to arbitrary graphs (digraphs) is using the notion of local resilience. The local resilience of a graph (digraph) G with respect to a property is the maximum number r such that G has the property even if we allow an adversary to remove an r-fraction of (in- and out-going) edges touching each vertex. The theorems of Dirac and Ghouila-Houri state that the local resilience of the complete graph and digraph with respect to Hamiltonicity is 1/2. Recently, this statements have been generalized to random settings. Lee and Sudakov (2012) proved that the local resilience of a random graph with edge probability p = ω (log n/n) with respect to Hamiltonicity is 1/2 ± o(1). For random directed graphs, Hefetz, Steger and Sudakov (2014+) proved an analogue statement, but only for edge probability . In this paper we significantly improve their result to p = ω (log8 n/n), which is optimal up to the polylogarithmic factor.