Whether local algorithms can compute constant approximations of NP-hard problems is of both practical and theoretical interest. So far, no algorithms achieving this goal are known, as either the approximation ratio or the running time exceed O(1), or the nodes are provided with non-trivial additional information. In this paper, we present the first distributed algorithm approximating a minimum dominating set on a planar graph within a constant factor in constant time. Moreover, the nodes do not need any additional information.

We are given a set V of autonomous agents (e.g. the computers of a distributed system) that are connected to each other by a graph G = (V, E) (e.g. by a communication network connecting the agents). Assume that all agents have a unique ID between 1 and N for a parameter N ≥ |V | and that each agent knows its ID as well as the IDs of its neighbors in G. Based on this limited information, every agent v must autonomously compute a set of colors Sv ⊆ C such that the color sets Su and Sv of adjacent agents u and v are disjoint. We prove that there is a deterministic algorithm that uses a total of |C | = O( ∆ 2 log(N)/ε 2) colors such that for every node v of G (i.e., for every agent), we have |Sv | ≥ |C|·(1−ε)/(δv+1), where δv is the degree of v and where ∆ is the maximum degree of G. For N = Ω( ∆ 2 log ∆), Ω( ∆ 2 + log log N) colors are necessary even to assign at least one color to every node (i.e., to compute a standard vertex coloring). Using randomization, it is possible to assign an (1 − ε)/(δ + 1)-fraction of all colors to every node of degree δ using only O( ∆ log |V |/ε 2) colors w.h.p. We show that this is asymptotically almost optimal. For graphs with maximum degree ∆ = Ω(log |V |), Ω( ∆ log |V | / log log |V |) colors are needed in expectation, even to compute a valid coloring. The described multicoloring problem has direct applications in the context of wireless ad hoc and

c t i v it y e p o r t 2009 Table of contents 1. Team.................................................................................... 1

c t i v it y e p o r t 2008 Table of contents 1. Team.................................................................................... 1

We present tradeoffs between time complexity t, bit complexity b, and message complexity m. Two communication parties can exchange Θ(m log(tb/m 2) + b) bits of information for m < √ bt and Θ(b) for m ≥ √ bt. This allows to derive lower bounds on the time complexity for distributed algorithms as we demonstrate for the MIS and the coloring problems. We reduce the bit-complexity of the state-of-the art O(∆) coloring algorithm without changing its time and message complexity. We also give techniques for several problems that require a time increase of t c (for an arbitrary constant c) to cut both bit and message complexity by Ω(log t). This improves on the traditional time-coding technique which does not allow to cut message complexity. 1

This work studies decision problems from the perspective of nondeterministic distributed algorithms. For a yes-instance there must exist a proof that can be verified with a distributed algorithm: all nodes must accept a valid proof, and at least one node must reject an invalid proof. We focus on locally checkable proofs that can be verified with a constanttime distributed algorithm. For example, it is easy to prove that a graph is bipartite: the locally checkable proof gives a 2-colouring of the graph, which only takes 1 bit per node. However, it is more difficult to prove that a graph is not bipartite—it turns out that any locally checkable proof requires Ω(log n) bits per node. In this work we classify graph problems according to their local proof complexity, i.e., how many bits per node are needed in a locally checkable proof. We establish tight or near-tight results for classical graph properties such as the chromatic number. We show that the proof complexities form a natural hierarchy of complexity classes: for many classical graph problems, the proof complexity is either 0, Θ(1), Θ(log n), or poly(n) bits per node. Among the most difficult graph properties are symmetric graphs, which require Ω(n 2) bits per node, and non-3-colourable graphs, which require Ω(n 2 / log n) bits per node—any pure graph property admits a trivial proof of size O(n 2).

Numerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and ( ∆ + 1)-coloring algorithms by Barenboim and Elkin [6], by Kuhn [22], and by Panconesi and Srinivasan [34], as well as the O(∆2)-coloring algorithm by Linial [28]. Unfortunately, most known local algorithms (including, in particular, the aforementioned algorithms) are non-uniform, that is, they assume that all nodes know good estimations of one or more global parameters of the network, e.g., the maximum degree ∆ or the number of nodes n. This paper provides a rather general method for transforming a non-uniform local algorithm into a uniform one. Furthermore, the resulting algorithm enjoys the same asymptotic running time as the original non-uniform algorithm. Our method applies to a wide family of both deterministic and randomized algorithms. Specifically, it applies to almost all of the state of the art non-uniform algorithms regarding MIS and Maximal Matching, as well as to many results concerning the coloring problem. (In particular, it applies to all aforementioned algorithms.) To obtain our transformations we introduce a new distributed tool called pruning algorithms, which we believe may be of independent interest.