Abstract In the stochastic population protocol model, we are given a connected graph with n nodes, and in every time step, a scheduler samples an edge of the graph uniformly at random and the nodes connected by this edge interact. A fundamental task in this model is stable leader election, in which all nodes start in an identical state and the aim is to reach a configuration in which (1) exactly one node is elected as leader and (2) this node remains as the unique leader no matter what sequence of interactions follows. On cliques, the complexity of this problem has recently been settled: time-optimal protocols stabilize in $$\Theta (n \log n)$$ Θ ( n log n ) expected steps using $$\Theta (\log \log n)$$ Θ ( log log n ) states, whereas protocols that use O(1) states require $$\Theta (n^2)$$ Θ ( n 2 ) expected steps. In this work, we investigate the complexity of stable leader election on graphs. We provide the first non-trivial time lower bounds on general graphs, showing that, when moving beyond cliques, the complexity of stable leader election can range from O(1) to $$\Theta (n^3)$$ Θ ( n 3 ) expected steps. We describe a protocol that is time-optimal on many graph families, but uses polynomially-many states. In contrast, we give a near-time-optimal protocol that uses only $$O(\log ^2n)$$ O ( log 2 n ) states that is at most a factor $$O(\log n)$$ O ( log n ) slower. Finally, we observe that for many graphs the constant-state protocol of Beauquier et al. [OPODIS 2013] is at most a factor $$O(n \log n)$$ O ( n log n ) slower than the fast polynomial-state protocol, and among constant-state protocols, this protocol has near-optimal average case complexity on dense random graphs.

Abstract The weighted vertex cover problem revolves around selecting a subset of vertices that covers a target edge set while minimizing the total cost of the selected vertices. We consider a variant of this classic optimization problem where the target edge set is not fully known; rather, it is characterized by a probability distribution. Adhering to the model of two-stage stochastic optimization, the execution is divided into two stages. In the first stage, the decision maker selects a vertex subset based on the probabilistic forecast of the target edge set. In the second stage, the target edge set is revealed, and the decision maker can augment the initial vertex subset with additional vertices to ensure coverage; however, this augmentation is more expensive due to increased vertex costs. This paper initiates the study of the two-stage stochastic vertex cover problem in the realm of distributed graph algorithms, where the decision-making process is distributed among the graph’s vertices. We consider two known stochastic optimization variants: the independent sampling model, where the edges in the target set are drawn independently from some probability distribution; and the finite scenario model, where the probability distribution over the target edge set is provided explicitly. For both variants, we devise efficient distributed algorithms based on a novel adaptation of the distributed primal-dual technique to linear programs resulting from the stochastic optimization problems’ relaxation.

Abstract In this paper we present efficient distributed algorithms for classical symmetry breaking problems, maximal independent sets (MIS) and ruling sets, in power graphs. We work in the standard CONGEST model of distributed message passing, where the communication network is abstracted as a graph G. Typically, the problem instance in CONGEST is identical to the communication network G, that is, we perform the symmetry breaking in G. In this work, we consider a setting where the problem instance corresponds to a power graph $$G^k$$ G k , where each node of the communication network G is connected to all of its k-hop neighbors. A $$\beta $$ β -ruling set is a set of non-adjacent nodes such that each node in G has a ruling neighbor within $$\beta $$ β hops; a natural generalization of an MIS. On top of being a natural family of problems, ruling sets (in power graphs) are well-motivated through their applications in the powerful shattering framework [BEPS JACM’16, Ghaffari SODA’19] (and others). We present randomized algorithms for computing maximal independent sets and ruling sets of $$G^k$$ G k in essentially the same time as they can be computed in G. Our main contribution is a deterministic $${{\,\textrm{poly}\,}}(k,\log n)$$ poly ( k , log n ) time algorithm for computing k-ruling sets of $$G^k$$ G k , which (for k > 1) improves exponentially on the current state-of-the-art runtimes. Our main technical ingredient for this result is a deterministic sparsification procedure which may be of independent interest. We also revisit the shattering algorithm for MIS [BEPS JACM’16] and present different approaches for the post-shattering phase. Our solutions are algorithmically and analytically simpler (also in the LOCAL model) than existing solutions and obtain the same runtime as [Ghaffari SODA’16].