with loops (undirected cycles). The algorithm is an exact inference algorithm for singly connected networks -the beliefs converge to the cor rect marginals in a number of iterations equal to the diameter of the graph.1 However, as Pearl noted, the same algorithm will not give the correct beliefs for mul

A complete partition of a graph is a partition of the vertex set such that any two classes are connected by an edge. We consider the problem of finding a complete partition maximizing the number of classes. This relates to clustering into the greatest number of groups so as to minimize the diameter

of a single node take roughly cubic time in the number n of nodes in the input graph. It is open whether these problems admit truly subcubic algorithms, i.e. algorithms with running time Õ(n3−δ) for some constant δ> 01. We relate the complexity of the mentioned centrality problems to two classical

We revisit the problem of distributed k-selection where, given a general connected graph of diameter D consisting of n nodes in which each node holds a numeric element, the goal is to determine the k th smallest of these elements. In our model, there is no imposed relation between the magnitude

We study the problem of minimizing the diameter of a graph by adding k shortcut edges, for speeding up communication in an existing network design. We develop constant-factor approximation algorithms for different variations of this problem. We also show how to improve the approximation ratios

that the transmission ranges of the stations ensure the communication between any pair of stations in at most h hops. Two main issues related to this problem are considered in this paper: the trade-off between the power consumption and the number of hops; the computational complexity of the MIN d D h-RANGE ASSIGNMENT

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We consider the problem of finding for a given weighted graph a minimum cost spanning tree whose diameter does not exceed a specified upper bound. This problem is NP-hard and has several applications, e.g. when designing communication networks and quality of service is of concern. We model

, by employing a reduction from the DOMINATING SET problem. We prove that the problem remains NP-hard even for forests and trees, but in this case we present approximation algorithms with worst-case bounds 3 (for even D) and 6 (for odd D). A closely related problem of finding a minimum number of edges

. The presented algorithm finds an O(log n) approximation in Õ(D+√n) rounds, where D is the network diameter and n is the number of nodes. MCDS is a classical NP-hard problem and the achieved approximation factor O(log n) is known to be optimal up to a constant factor, unless P = NP. Furthermore, the Õ(D +√n