10,434 research outputs found

    Approximating the Expansion Profile and Almost Optimal Local Graph Clustering

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    Spectral partitioning is a simple, nearly-linear time, algorithm to find sparse cuts, and the Cheeger inequalities provide a worst-case guarantee for the quality of the approximation found by the algorithm. Local graph partitioning algorithms [ST08,ACL06,AP09] run in time that is nearly linear in the size of the output set, and their approximation guarantee is worse than the guarantee provided by the Cheeger inequalities by a polylogarithmic logΩ(1)n\log^{\Omega(1)} n factor. It has been a long standing open problem to design a local graph clustering algorithm with an approximation guarantee close to the guarantee of the Cheeger inequalities and with a running time nearly linear in the size of the output. In this paper we solve this problem; we design an algorithm with the same guarantee (up to a constant factor) as the Cheeger inequality, that runs in time slightly super linear in the size of the output. This is the first sublinear (in the size of the input) time algorithm with almost the same guarantee as the Cheeger's inequality. As a byproduct of our results, we prove a bicriteria approximation algorithm for the expansion profile of any graph. Let ϕ(γ)=minμ(S)γϕ(S)\phi(\gamma) = \min_{\mu(S) \leq \gamma}\phi(S). There is a polynomial time algorithm that, for any γ,ϵ>0\gamma,\epsilon>0, finds a set SS of measure μ(S)2γ1+ϵ\mu(S)\leq 2\gamma^{1+\epsilon}, and expansion ϕ(S)2ϕ(γ)/ϵ\phi(S)\leq \sqrt{2\phi(\gamma)/\epsilon}. Our proof techniques also provide a simpler proof of the structural result of Arora, Barak, Steurer [ABS10], that can be applied to irregular graphs. Our main technical tool is that for any set SS of vertices of a graph, a lazy tt-step random walk started from a randomly chosen vertex of SS, will remain entirely inside SS with probability at least (1ϕ(S)/2)t(1-\phi(S)/2)^t. This itself provides a new lower bound to the uniform mixing time of any finite states reversible markov chain

    Maximum flow is approximable by deterministic constant-time algorithm in sparse networks

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    We show a deterministic constant-time parallel algorithm for finding an almost maximum flow in multisource-multitarget networks with bounded degrees and bounded edge capacities. As a consequence, we show that the value of the maximum flow over the number of nodes is a testable parameter on these networks.Comment: 8 page
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