We study partial and budgeted versions of the well studied connected dominating set problem. In the partial connected dominating set problem (Pcds), we are given an undirected graph G = (V, E) and an integer n , and the goal is to find a minimum subset of vertices that induces a connected subgraph of G and dominates at least n vertices. We obtain the first polynomial time algorithm with an O(ln ∆) approximation factor for this problem, thereby significantly extending the results of Guha and Khuller (Algorithmica, Vol. 20(4), Pages 374-387, 1998) for the connected dominating set problem. We note that none of the methods developed earlier can be applied directly to solve this problem. In the budgeted connected dominating set problem (Bcds), there is a budget on the number of vertices we can select, and the goal is to dominate as many vertices as possible. We obtain a 1 13(1 − 1 e ) approximation algorithm for this problem. Finally, we show that our techniques extend to a more general setting where the profit function associated with a subset of vertices is a "special" submodular function. This generalization captures the connected dominating set problem with capacities and/or weighted profits as special cases. This implies a O(ln q) approximation (where q denotes the quota) and an O(1) approximation algorithms for the partial and budgeted versions of these problems. While the algorithms are simple, the results make a surprising use of the greedy set cover framework in defining a useful profit function.
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Given a social network, can we quickly 'zoom-out' of the graph? Is there a smaller equivalent representation of the graph that preserves its propagation characteristics? Can we group nodes together based on their influence properties? These are important problems with applications to influence analysis, epidemiology and viral marketing applications.In this paper, we first formulate a novel Graph Coarsening Problem to find a succinct representation of any graph while preserving key characteristics for diffusion processes on that graph. We then provide a fast and effective near-linear-time (in nodes and edges) algorithm coarseNet for the same. Using extensive experiments on multiple real datasets, we demonstrate the quality and scalability of coarseNet, enabling us to reduce the graph by 90% in some cases without much loss of information. Finally we also show how our method can help in diverse applications like influence maximization and detecting patterns of propagation at the level of automatically created groups on real cascade data.
The coflow scheduling problem has emerged as a popular abstraction in the last few years to study data communication problems within a data center [6]. In this basic framework, each coflow has a set of communication demands and the goal is to schedule many coflows in a manner that minimizes the total weighted completion time. A coflow is said to complete when all its communication needs are met. This problem has been extremely well studied for the case of complete bipartite graphs that model a data center with full bisection bandwidth and several approximation algorithms and effective heuristics have been proposed recently [1,2,29].In this work, we study a slightly different model of coflow scheduling in general graphs (to capture traffic between data centers [15,29]) and develop practical and efficient approximation algorithms for it. Our main result is a randomized 2 approximation algorithm for the single path and free path model, significantly improving prior work. In addition, we demonstrate via extensive experiments that the algorithm is practical, easy to implement and performs well in practice.
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