Final Report Summary - ACDC (Algorithms and Complexity of Highly Decentralized Computations) The goal of the ACDC project was to obtain a basic understanding of the fundamental possibilities and impossibilities of large, decentralized distributed systems. In an increasingly networked world, such systems will become more and more important. The project concentrated on three main aspects. Typically, the computation of a node in a network mostly depends on the states of other close-by nodes. A core objective therefore was to better understand to what extent a global goal can be achieved based on only local information. Further, the project intended to improve our understanding of distributed computations in wireless and dynamic networks, two scenarios where the issues of locality and decentralized organization are particularly important. In the project, we made progress on many aspects regarding the complexity of distributed computations in large networks. In the following, we only briefly discuss the most important results.In the context of dynamic and wireless networks, we most importantly improved our understanding of the cost of disseminating information in the network. In particular, we studied wireless radio networks, where the topology can change dynamically over time in a worst-case manner. We established possible trade-offs between the time for spreading information and the possible dynamic behavior of the network. Our research on information dissemination in dynamic networks also lead to a series of works where established some new bounds on the cost of information dissemination in classic, static networks. In particular, we gave formal and tight mathematical trade-offs between how well a network is connected and how fast information can be spread in the network. This work also lead to some purely graph-theoretic results and in particular to a precise understanding of how the vertex connectivity of a graph behaves under random sampling of the vertices.The most important results of the project deal with the locality of distributed computations and there with the following 30-year old fundamental open problem. For many of the basic and standard graph problems, we have very efficient randomized algorithms that run in time that is only logarithmic or polylogarithmic in the number of nodes n of the network. In many of these cases, the best deterministic algorithms are exponentially slower and it is an important open problem to understand whether such an exponential separation is inherent or whether all problems that can be solved efficiently with randomization can also be solved efficiently deterministically. We made significant progress in understanding the role of randomization for such problems in several ways. We developed a complexity-theoretic framework that allows to analyze the separation between randomized and deterministic algorithms in a mathematically clean and systematic way. This in particular allowed to identify some simple so-called complete problems. If any of these problems could be solved efficiently by deterministic algorithms, then all problems that have efficient randomized distributed algorithms also have efficient deterministic distributed algorithms.A particularly important problem for which currently such an exponential separation exists is the distributed coloring problem, where the goal is to color each node of a network with a color such that no two neighbors have the same color and such that the total number of colors is a small as possible. For a special case of this problem, where instead of the nodes, we need to color the edges (i.e. the communication links of the network), we were able to give the first efficient deterministic algorithm and thus solve a long-standing open problem. We also developed a general technique to derandomize randomized distributed algorithms, which also allowed to obtain the first efficient deterministic distributed algorithms on some other fundamental graph problems. Further, for a somewhat weakened communication model, we gave one of the first time lower bounds for the standard distributed coloring problem.
