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Streaming Lower Bounds for Approximating MAXCUT
, 2014
"... We consider the problem of estimating the value of max cut in a graph in the streaming model of computation. At one extreme, there is a trivial 2approximation for this problem that uses only O(log n) space, namely, count the number of edges and output half of this value as the estimate for max cut ..."
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We consider the problem of estimating the value of max cut in a graph in the streaming model of computation. At one extreme, there is a trivial 2approximation for this problem that uses only O(log n) space, namely, count the number of edges and output half of this value as the estimate for max cut value. On the other extreme, if one allows Õ(n) space, then a nearoptimal solution to the max cut value can be obtained by storing an Õ(n)size sparsifier that essentially preserves the max cut. An intriguing question is if polylogarithmic space suffices to obtain a nontrivial approximation to the maxcut value (that is, beating the factor 2). It was recently shown that the problem of estimating the size of a maximum matching in a graph admits a nontrivial approximation in polylogarithmic space. Our main result is that any streaming algorithm that breaks the 2approximation barrier requires Ω̃( n) space even if the edges of the input graph are presented in random order. Our result is obtained by exhibiting a distribution over graphs which are either bipartite or 12far from being bipartite, and establishing that Ω̃( n) space is necessary to differentiate between these two cases. Thus as a direct corollary we obtain that
Vertex and Hyperedge Connectivity in Dynamic Graph Streams
"... A growing body of work addresses the challenge of processing dynamic graph streams: a graph is defined by a sequence of edge insertions and deletions and the goal is to construct synopses and compute properties of the graph while using only limited memory. Linear sketches have proved to be a powerfu ..."
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A growing body of work addresses the challenge of processing dynamic graph streams: a graph is defined by a sequence of edge insertions and deletions and the goal is to construct synopses and compute properties of the graph while using only limited memory. Linear sketches have proved to be a powerful technique in this model and can also be used to minimize communication in distributed graph processing. We present the first linear sketches for estimating vertex connectivity and constructing hypergraph sparsifiers. Vertex connectivity exhibits markedly different combinatorial structure than edge connectivity and appears to be harder to estimate in the dynamic graph stream model. Our hypergraph result generalizes the work of Ahn et al. (PODS 2012) on graph sparsification and has the added benefit of significantly simplifying the previous results. One of the main ideas is related to the problem of reconstructing subgraphs that satisfy a specific sparsity property. We introduce a more general notion of graph degeneracy and extend the graph reconstruction result of Becker et al. (IPDPS 2011). 1
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, 2014
"... The emergence of massive datasets has led to the rise of new computational paradigms where computation is limited. In the streaming model the input graph is presented to the algorithm as a stream of edges which is prohibitively large to be stored in its entirety (i.e., the algorithm’s space complexi ..."
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The emergence of massive datasets has led to the rise of new computational paradigms where computation is limited. In the streaming model the input graph is presented to the algorithm as a stream of edges which is prohibitively large to be stored in its entirety (i.e., the algorithm’s space complexity must be small relative to the stream size). After reading the stream, the algorithm should report a solution to a predetermined problem on the graph. In the sketching model, the input graph is summarized into a socalled sketch, which is short yet suffices for further processing without access to the original input. Cuts in graphs is a classical topic of both theoretical and practical interest, studied extensively for more than half a century. A graph cut is a partition of the vertices to two disjoint sets, and the value of the cut is the number of edges (or their total weight in case the graph is weighted) with one endpoint in each part of the partition. This definition can be extended to runiform hypergraphs, in which case hyperedges are sets of r vertices, and a hyperedge belongs to the cut if it intersects both parts of the vertex bipartition. We first address a natural question, whether the the value of the maximum cut in a graph admits