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].

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 P is the maximum number r such that G has the property P 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 p = ω (log n/ √ n). In this paper we significantly improve their result to p = ω log 8 n/n , which is optimal up to the polylogarithmic factor.A Hamilton cycle in a graph or a directed graph is a cycle that passes through all the vertices of the graph exactly once, and a graph is Hamiltonian if it contains a Hamilton cycle. Hamiltonicity is one of the central notions in graph theory, and has been intensively studied by numerous researchers. It is well known that the problem of whether a given graph contains a Hamilton cycle is N P-complete. In fact, Hamiltonicity was one of Karp's 21 N P-complete problems [12].Since one can not hope for a general classification of Hamiltonian graphs, as a consequence of Karp's result, there is a large interest in deriving properties that are sufficient for Hamiltonicity. A classic result by Dirac from 1952 [7] states that every graph on n ≥ 3 vertices with minimum degree at least n/2 is Hamiltonian. This result is tight as the complete bipartite graph with parts of sizes that differ by one, K m,m+1 , is not Hamiltonian. Note that this theorem answers the following question: Starting with the complete graph on n vertices K n , what is the maximal integer ∆ such that for any subgraph H of K n with maximum degree ∆, the graph K n − H obtained by deleting the edges of H from K n is Hamiltonian? This question not only asks for a sufficient condition for a graph to be Hamiltonian, it also asks for a quantification for the "local robustness" of the complete graph with respect to Hamiltonicity.A natural generalization of this question is to replace the complete graph with some arbitrary base graph. Recently, questions of this type have drawn a lot of attention under...

We consider collaborative graph exploration with a set of k agents. All agents start at a common vertex of an initially unknown graph and need to collectively visit all other vertices. We assume agents are deterministic, vertices are distinguishable, moves are simultaneous, and we allow agents to communicate globally. For this setting, we give the first non-trivial lower bounds that bridge the gap between small (k ≤ √ n) and large (k ≥ n) teams of agents. Remarkably, our bounds tightly connect to existing results in both domains. First, we significantly extend a lower bound of Ω(log k/ log log k) by Dynia et al. on the competitive ratio of a collaborative tree exploration strategy to the range k ≤ n log c n for any c ∈ N. Second, we provide a tight lower bound on the number of agents needed for any competitive exploration algorithm. In particular, we show that any collaborative tree exploration algorithm with k = Dn 1+o(1) agents has a competitive ratio of ω(1), while Dereniowski et al. gave an algorithm with k = Dn 1+ε agents and competitive ratio O(1), for any ε > 0 and with D denoting the diameter of the graph. Lastly, we show that, for any exploration algorithm using k = n agents, there exist trees of arbitrarily large height D that require Ω(D 2 ) rounds, and we provide a simple algorithm that matches this bound for all trees.

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(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].

A classic result of Erdős and Pósa states that any graph either contains k vertexdisjoint cycles or can be made acyclic by deleting at most O(k log k) vertices. Birmelé, Bondy, and Reed (2007) raised the following more general question: given numbers l and k, what is the optimal function f (l, k) such that every graph G either contains k vertex-disjoint cycles of length at least l or contains a set X of f (l, k) vertices that meets all cycles of length at least l? In this paper, we answer that question by proving that f (l, k) = Θ(kl + k log k). As a corollary, the tree-width of any graph G that does not contain k vertexdisjoint cycles of length at least l is of order O(kl + k log k). This is also optimal up to constant factors and answers another question of Birmelé, Bondy, and Reed (2007).