No abstract

We define a generalization of local distributed graph problems to (synchronous roundbased) dynamic networks and present a framework for developing algorithms for these problems. The algorithms should satisfy non-trivial guarantees in every round. The guarantees should be stronger the more stable the graph has been during the last few rounds and coincide with the definition of the static graph problem if no topological change appeared recently. Moreover, if only a constant neighborhood around some part of the graph is stable during an interval, the algorithms should quickly converge to a solution for this part of the graph that remains unchanged throughout the interval.We demonstrate our generic framework with two classic distributed graph problems, namely (degree+1)-vertex coloring and maximal independent set (MIS). To illustrate the given guarantees consider the vertex coloring problem: Any conflict between two nodes caused by a newly inserted edge is resolved within T = O(log n) rounds. During this conflict resolving both nodes always output colors that are not in conflict with their respective 'old' neighbors. The largest color that a node is allowed to output is determined by the number of distinct neighbors that it has seen in the last T rounds.

One of the fundamental open problems in the area of distributed graph algorithms is the question of whether randomization is needed for efficient symmetry breaking. While there are fast, poly log n-time randomized distributed algorithms for all of the classic symmetry breaking problems, for many of them, the best deterministic algorithms are almost exponentially slower. The following basic local splitting problem, which is known as the weak splitting problem takes a central role in this context: Each node of a graph G = (V, E) has to be colored red or blue such that each node of sufficiently large degree has at least one node of each color among its neighbors. Ghaffari, Kuhn, and Maus [STOC '17] showed that this seemingly simple problem is complete w.r.t. the above fundamental open question in the following sense: If there is an efficient poly log n-time determinstic distributed algorithm for weak splitting, then there is such an algorithm for all locally checkable graph problems for which an efficient randomized algorithm exists. In this paper, we investigate the distributed complexity of weak splitting and some closely related problems and we in particular obtain the following results:• We obtain efficient algorithms for special cases of weak splitting, where the graph is nearly regular. In particular, we show that if δ and ∆ are the minimum and maximum degrees of G and if δ = Ω(log n), weak splitting can be solved deterministically in time O ∆ δ · poly(log n) . Further, if δ = Ω(log log n) and ∆ ≤ 2 εδ , there is a randomized algorithm with time complexity O ∆ δ · poly(log log n) . • We prove that the following two related problems are also complete in the same sense:(I) Color the nodes of a graph with C ≤ poly log n colors such that each node with a sufficiently large polylogarithmic degree has at least 2 log n colors among its neighbors, and (II) Color the nodes with a large constant number of colors so that for each node of a sufficiently large at least logarithmic degree d(v), the number of neighbors of each color is at most (1 − ε)d(v) for some constant ε > 0.