In the Byzantine agreement problem, n nodes with possibly different input values aim to reach agreement on a common value in the presence of t < n/3 Byzantine nodes which represent arbitrary failures in the system. This paper introduces a generalization of Byzantine agreement, where the input values of the nodes are preference rankings of three or more candidates. We show that consensus on preferences, which is an important question in social choice theory, complements already known results from Byzantine agreement. In addition preferential voting raises new questions about how to approximate consensus vectors. We propose a deterministic algorithm to solve Byzantine agreement on rankings under a generalized validity condition, which we call Pareto -Validity. These results are then extended by considering a special voting rule which chooses the Kemeny median as the consensus vector. For this rule, we derive a lower bound on the approximation ratio of the Kemeny median that can be guaranteed by any deterministic algorithm. We then provide an algorithm matching this lower bound. To our knowledge, this is the first non-trivial multi-dimensional approach which can tolerate a constant fraction of Byzantine nodes.
In this paper, we study k-Way Min-cost Perfect Matching with Delays -the k-MPMD problem. This problem considers a metric space with n nodes. Requests arrive at these nodes in an online fashion. The task is to match these requests into sets of exactly k, such that space and time cost of all matched requests are minimized. The notion of the space cost requires a definition of an underlying metric space that gives distances of subsets of k elements. For k > 2, the task of finding a suitable metric space is at the core of our problem: We show that for some known generalizations to k = 3 points, such as the 2-metric [23] and the D-metric [43], there exists no competitive randomized algorithm for the 3-MPMD problem. The G-metrics [39] are defined for 3 points and allows for a competitive algorithm for the 3-MPMD problem. For k > 3 points, there exist two generalizations of the G-metrics known as n-and K-metrics [4,31]. We show that neither the n-metrics nor the K-metrics can be used for the k-MPMD problem. On the positive side, we introduce the H-metrics, the first metrics to allow for a solution of the k-MPMD problem for all k. In order to devise an online algorithm for the k-MPMD problem on the H-metrics, we embed the H-metric into trees with an O(log n) distortion. Based on this embedding result, we extend the algorithm proposed by Azar et al. [7] and achieve a competitive ratio of O(log n) for the k-MPMD problem.
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