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Sketching cuts in graphs and hypergraphs
, 2014
"... Sketching and streaming algorithms are in the forefront of current research directions for cut problems in graphs. In the streaming model, we show that (1 − ε)approximation for MaxCut must use n1−O(ε) space; moreover, beating 4/5approximation requires polynomial space. For the sketching model, we ..."
Abstract

Cited by 4 (1 self)
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Sketching and streaming algorithms are in the forefront of current research directions for cut problems in graphs. In the streaming model, we show that (1 − ε)approximation for MaxCut must use n1−O(ε) space; moreover, beating 4/5approximation requires polynomial space. For the sketching model, we show that runiform hypergraphs admit a (1 + ε)cutsparsifier (i.e., a weighted subhypergraph that approximately preserves all the cuts) with O(ε−2n(r + log n)) edges. We also make first steps towards sketching general CSPs (Constraint Satisfaction Problems). 1
Approximation algorithms for hypergraph small set expansion and small set vertex expansion
 CoRR
"... The expansion of a hypergraph, a natural extension of the notion of expansion in graphs, is defined as the minimum over all cuts in the hypergraph of the ratio of the number of the hyperedges cut to the size of the smaller side of the cut. We study the Hypergraph Small Set Expansion problem, which ..."
Abstract

Cited by 3 (1 self)
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The expansion of a hypergraph, a natural extension of the notion of expansion in graphs, is defined as the minimum over all cuts in the hypergraph of the ratio of the number of the hyperedges cut to the size of the smaller side of the cut. We study the Hypergraph Small Set Expansion problem, which, for a parameter δ ∈ (0, 1/2], asks to compute the cut having the least expansion while having at most δ fraction of the vertices on the smaller side of the cut. We present two algorithms. Our first algorithm gives an Õ(δ−1 log n) approximation. The second algorithm finds a set with expansion Õ(δ−1( dmaxr−1 log r φ ∗ + φ∗)) in a r–uniform hypergraph with maximum degree dmax (where φ ∗ is the expansion of the optimal solution). Using these results, we also obtain algorithms for the Small Set Vertex Expansion problem: we get an Õ(δ−1 log n) approximation algorithm and an algorithm that finds a set with vertex expansion O δ−1 φV log dmax + δ
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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 ..."
Abstract
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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