11 research outputs found

    Detecting and Characterizing Small Dense Bipartite-like Subgraphs by the Bipartiteness Ratio Measure

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    We study the problem of finding and characterizing subgraphs with small \textit{bipartiteness ratio}. We give a bicriteria approximation algorithm \verb|SwpDB| such that if there exists a subset SS of volume at most kk and bipartiteness ratio θ\theta, then for any 0<ϵ<1/20<\epsilon<1/2, it finds a set SS' of volume at most 2k1+ϵ2k^{1+\epsilon} and bipartiteness ratio at most 4θ/ϵ4\sqrt{\theta/\epsilon}. By combining a truncation operation, we give a local algorithm \verb|LocDB|, which has asymptotically the same approximation guarantee as the algorithm \verb|SwpDB| on both the volume and bipartiteness ratio of the output set, and runs in time O(ϵ2θ2k1+ϵln3k)O(\epsilon^2\theta^{-2}k^{1+\epsilon}\ln^3k), independent of the size of the graph. Finally, we give a spectral characterization of the small dense bipartite-like subgraphs by using the kkth \textit{largest} eigenvalue of the Laplacian of the graph.Comment: 17 pages; ISAAC 201

    Lower Bounds on Expansions of Graph Powers

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    Given a lazy regular graph G, we prove that the expansion of G^t is at least sqrt(t) times the expansion of G. This bound is tight and can be generalized to small set expansion. We show some applications of this result

    Testing Small Set Expansion in General Graphs

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    We consider the problem of testing small set expansion for general graphs. A graph GG is a (k,ϕ)(k,\phi)-expander if every subset of volume at most kk has conductance at least ϕ\phi. Small set expansion has recently received significant attention due to its close connection to the unique games conjecture, the local graph partitioning algorithms and locally testable codes. We give testers with two-sided error and one-sided error in the adjacency list model that allows degree and neighbor queries to the oracle of the input graph. The testers take as input an nn-vertex graph GG, a volume bound kk, an expansion bound ϕ\phi and a distance parameter ε>0\varepsilon>0. For the two-sided error tester, with probability at least 2/32/3, it accepts the graph if it is a (k,ϕ)(k,\phi)-expander and rejects the graph if it is ε\varepsilon-far from any (k,ϕ)(k^*,\phi^*)-expander, where k=Θ(kε)k^*=\Theta(k\varepsilon) and ϕ=Θ(ϕ4min{log(4m/k),logn}(lnk))\phi^*=\Theta(\frac{\phi^4}{\min\{\log(4m/k),\log n\}\cdot(\ln k)}). The query complexity and running time of the tester are O~(mϕ4ε2)\widetilde{O}(\sqrt{m}\phi^{-4}\varepsilon^{-2}), where mm is the number of edges of the graph. For the one-sided error tester, it accepts every (k,ϕ)(k,\phi)-expander, and with probability at least 2/32/3, rejects every graph that is ε\varepsilon-far from (k,ϕ)(k^*,\phi^*)-expander, where k=O(k1ξ)k^*=O(k^{1-\xi}) and ϕ=O(ξϕ2)\phi^*=O(\xi\phi^2) for any 0<ξ<10<\xi<1. The query complexity and running time of this tester are O~(nε3+kεϕ4)\widetilde{O}(\sqrt{\frac{n}{\varepsilon^3}}+\frac{k}{\varepsilon \phi^4}). We also give a two-sided error tester with smaller gap between ϕ\phi^* and ϕ\phi in the rotation map model that allows (neighbor, index) queries and degree queries.Comment: 23 pages; STACS 201
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