Symmetry breaking problems are among the most well studied in the field of distributed computing and yet the most fundamental questions about their complexity remain open. In this paper we work in the LOCAL model (where the input graph and underlying distributed network are identical) and study the randomized complexity of four fundamental symmetry breaking problems on graphs: computing MISs (maximal independent sets), maximal matchings, vertex colorings, and ruling sets. A small sample of our results includes• An MIS algorithm running in O(log 2 ∆ + 2 O( √ log log n) ) time, where ∆ is the maximum degree. This is the first MIS algorithm to improve on the 1986 algorithms of Luby and Alon, Babai, and Itai, when log n ∆ 2 √ log n , and comes close to the Ω(log ∆) lower bound of Kuhn, Moscibroda, and Wattenhofer.• A maximal matching algorithm running in O(log ∆ + log 4 log n) time. This is the first significant improvement to the 1986 algorithm of Israeli and Itai. Moreover, its dependence on ∆ is provably optimal. • A (∆ + 1)-coloring algorithm requiring O(log ∆ + 2 O( IntroductionBreaking symmetry is one of the central themes in the theory of distributed computing. At initialization the nodes of a distributed system are assumed to be in the same state, possibly with distinct * A preliminary version of this paper appeared in the

We study the distributed maximal independent set (henceforth, MIS) problem on sparse graphs. Currently, there are known algorithms with a sublogarithmic running time for this problem on oriented trees and graphs of bounded degrees. We devise the first sublogarithmic algorithm for computing MIS on graphs of bounded arboricity. This is a large family of graphs that includes graphs of bounded degree, planar graphs, graphs of bounded genus, graphs of bounded treewidth, graphs that exclude a fixed minor, and many other graphs. We also devise efficient algorithms for coloring graphs from these families.These results are achieved by the following technique that may be of independent interest. Our algorithm starts with computing a certain graph-theoretic structure, called NashWilliams forests-decomposition. Then this structure is used to compute the MIS or coloring. Our results demonstrate that this methodology is very powerful.Finally, we show nearly-tight lower bounds on the running time of any distributed algorithm for computing a forestsdecomposition.

The focus of this monograph is on symmetry breaking problems in the message-passing model of distributed computing. In this model a communication network is represented by a n-vertex graph G = (V, E), whose vertices host autonomous processors. The processors communicate over the edges of G in discrete rounds. The goal is to devise algorithms that use as few rounds as possible.A typical symmetry breaking problem is the problem of graph coloring. Denote by ∆ the maximum degree of G. While coloring G with ∆ + 1 colors is trivial in the centralized setting, the problem becomes much more challenging in the distributed one. One can also compromise on the number of colors, if this allows for more efficient algorithms. Other typical symmetry-breaking problems are the problems of computing a maximal independent set (MIS) and a maximal matching (MM). The study of these problems dates back to the very early days of distributed computing. The founding fathers of distributed computing (see [60,61,39,40,18,55,31,5]) laid firm foundations for the area of distributed symmetry breaking already in the eighties. In particualr, they showed that all these problems can be solved in randomized logarithmic time. Also, Linial [55] showed that an O(∆ 2 )-coloring can be solved very efficiently deterministically.However, fundamental questions were left open for decades. In particular, it is not known if the MIS or the (∆ + 1)-coloring can be solved in deterministic polylogarithmic time. Moreover, until recently it was not known if in deterministic polylogarithmic time one can color a graph with significantly less than ∆ 2 colors. Additionally, it was open (and still open to some extent) if one can have sublogarithmic randomized algorithms for the symmetry breaking problems.Recently, significant progress was achieved in the study of these questions. More efficient deterministic and randomized (∆ + 1)-coloring algorithms were achieved in [8,48,78,11]. Deterministic ∆ 1+o(1) -coloring algorithms with polylogarithmic running time were devised in [9]. Improved (and often sublogarithmic-time) randomized algorithms were devised in [47,78,11]. Drastically improved lower bounds were given in [50,52]. Wide families of graphs in which these problems are solvable much faster than on general graphs were identified in [49,51,30,77,7,9].The objective of our monograph is to cover most of these developments, and as a result to provide a treatise on theoretical foundations of distributed symmetry breaking in the message-passing model. We hope that our monograph will stimulate further progress in this exciting area.

The distributed (∆ + 1)-coloring problem is one of most fundamental and well-studied problems in Distributed Algorithms. Starting with the work of Cole and Vishkin in 86, there was a long line of gradually improving algorithms published. The current state-of-the-art running time is O(∆ log ∆+log * n), due to Kuhn and Wattenhofer, PODC'06. Linial (FOCS'87) has proved a lower bound of 1 2 log * n for the problem, and Szegedy and Vishwanathan (STOC'93) provided a heuristic argument that shows that algorithms from a wide family of locally iterative algorithms are unlikely to achieve running time smaller than Θ(∆ log ∆). We present a deterministic (∆+1)-coloring distributed algorithm with running time O(∆)+ 1 2 log * n. We also present a tradeoff between the running time and the number of colors , and devise an O(∆ · t)-coloring algorithm with running time O(∆/t+log * n), for any parameter t, 1 < t ≤ ∆ 1− , for an arbitrarily small constant , 0 < < 1. Our algorithm breaks the heuristic barrier of Szegedy and Vishwanathan, and achieves running time which is linear in the maximum degree ∆. On the other hand, the conjecture of Szegedy and Vishwanathan may still be true, as our algorithm is not from the family of locally iterative algorithms. On the way to this result we study a generalization of the notion of graph coloring, which is called defective coloring. In an m-defective p-coloring the vertices are colored with p colors so that each vertex has up to m neighbors with the same color. We show that an m-defective p-coloring with reasonably small m and p can be computed very efficiently. We also develop a technique to employ multiple defective colorings of various subgraphs of the original graph G for computing a (∆ + 1)-coloring of G. We believe that these techniques are of independent interest. *

We study the distributed maximal independent set (henceforth, MIS) problem on sparse graphs. Currently, there are known algorithms with a sublogarithmic running time for this problem on oriented trees and graphs of bounded degrees. We devise the first sublogarithmic algorithm for computing MIS on graphs of bounded arboricity. This is a large family of graphs that includes graphs of bounded degree, planar graphs, graphs of bounded genus, graphs of bounded treewidth, graphs that exclude a fixed minor, and many other graphs. We also devise efficient algorithms for coloring graphs from these families.These results are achieved by the following technique that may be of independent interest. Our algorithm starts with computing a certain graph-theoretic structure, called NashWilliams forests-decomposition. Then this structure is used to compute the MIS or coloring. Our results demonstrate that this methodology is very powerful.Finally, we show nearly-tight lower bounds on the running time of any distributed algorithm for computing a forestsdecomposition.