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We consider an information dissemination problem where the root of an undirected graph constantly updates its information and we wish to keep everyone in the graph as freshly informed about the root as possible. Our synchronous information spreading model uses telephone calls at each time step, in which any node can call at most one neighbor, thus inducing a matching over which information is transmitted at each step. These matchings that define the schedule over time ad infinitum are specified in a centralized manner. We introduce two problems in minimizing two natural objectives (Maximum and Average) of the latency of the root's information at all nodes in the network.First we show that it suffices to consider periodic finite schedules with a loss of at most a factor of two in each of these objectives. Next, by relating the maximum rooted latency problem to minimizing the maximum broadcast-time, we derive new approximation algorithms for this version. For the average rooted latency objective, such an approach introduces a necessary logarithmic factor overhead. We overcome this and give a constant-factor approximation for trees by devising near-optimal schedules that are locally periodic at each node and where each node has the same offset value in hearing fresh information from its parent. The average latency version motivates the problem of finding a spanning tree that minimizes a new Average Broadcast Time objective in graphs, which is interesting in its own right for future study.

The minimum spanning tree of a graph is a well-studied structure that is the basis of countless graph theoretic and optimization problem. We study the minimum spanning tree (MST) perturbation problem where the goal is to spend a fixed budget to increase the weight of edges in order to increase the weight of the MST as much as possible. Two popular models of perturbation are bulk and continuous. In the bulk model, the weight of any edge can be increased exactly once to some predetermined weight. In the continuous model, one can pay a fractional amount of cost to increase the weight of any edge by a proportional amount. Frederickson and Solis-Oba [12] have studied these two models and showed that bulk perturbation for MST is as hard as the k-cut problem while the continuous perturbation model is solvable in poly-time.In this paper, we study an intermediate unit perturbation variation of this problem where the weight of each edge can be increased many times but at an integral unit amount every time. We provide an (opt/2 − 1)approximation in polynomial time where opt is the optimal increase in the weight. We also study the associated dual targeted version of the problem where the goal is to increase the weight of the MST by a target amount while minimizing the cost of perturbation. We provide a 2-approximation for this variation. Furthermore we show that assuming the Small Set Expansion Hypothesis, both problems are hard to approximate. We also point out an error in the proof provided by Frederickson and Solis-Oba in [12] with regard to their solution to the continuous perturbation model. Although their algorithm is correct, their analysis is flawed. We provide a correct proof here.