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**1 - 5**of**5**### Detecting and Characterizing Small Dense Bipartite-like Subgraphs by the Bipartiteness Ratio Measure

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### NEW ROUNDING TECHNIQUES FOR THE DESIGN AND ANALYSIS OF APPROXIMATION ALGORITHMS

, 2014

"... We study two of the most central classical optimization problems, namely the Traveling Salesman problems and Graph Partitioning problems and develop new approximation algorithms for them. We introduce several new techniques for rounding a fractional solution of a continuous relaxation of these probl ..."

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We study two of the most central classical optimization problems, namely the Traveling Salesman problems and Graph Partitioning problems and develop new approximation algorithms for them. We introduce several new techniques for rounding a fractional solution of a continuous relaxation of these problems into near optimal integral solutions. The two most notable of those are the maximum entropy rounding by sampling method and a novel use of higher eigenvectors of graphs.

### Locally Finding and Testing Dense Bipartite-like Subgraphs

"... Abstract We study local algorithms for finding and testing dense bipartite-like subgraphs which characterize cyber-communities in the web ..."

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Abstract We study local algorithms for finding and testing dense bipartite-like subgraphs which characterize cyber-communities in the web

### Markov chain methods for small-set expansion

, 2012

"... Consider a finite irreducible Markov chain with invariant distribution π. We use the inner product induced by π and the associated heat operator to simplify and generalize some results related to graph partitioning and the small-set expansion problem. For example, Steurer showed a tight connection b ..."

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Consider a finite irreducible Markov chain with invariant distribution π. We use the inner product induced by π and the associated heat operator to simplify and generalize some results related to graph partitioning and the small-set expansion problem. For example, Steurer showed a tight connection between the number of small eigenvalues of a graph’s Laplacian and the expansion of small sets in that graph. We give a simplified proof which generalizes to the nonregular, directed case. This result implies an approximation algorithm for an “analytic” version of the Small-Set Expansion Problem, which, in turn, immediately gives an approximation algorithm for Small-Set Expansion. We also give a simpler proof of a lower bound on the probability that a random walk stays within a set; this result was used in some recent works on finding small sparse cuts. 1