4,806 research outputs found

    Interlacing Families IV: Bipartite Ramanujan Graphs of All Sizes

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    We prove that there exist bipartite Ramanujan graphs of every degree and every number of vertices. The proof is based on analyzing the expected characteristic polynomial of a union of random perfect matchings, and involves three ingredients: (1) a formula for the expected characteristic polynomial of the sum of a regular graph with a random permutation of another regular graph, (2) a proof that this expected polynomial is real rooted and that the family of polynomials considered in this sum is an interlacing family, and (3) strong bounds on the roots of the expected characteristic polynomial of a union of random perfect matchings, established using the framework of finite free convolutions we recently introduced

    Ramanujan Graphs in Polynomial Time

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    The recent work by Marcus, Spielman and Srivastava proves the existence of bipartite Ramanujan (multi)graphs of all degrees and all sizes. However, that paper did not provide a polynomial time algorithm to actually compute such graphs. Here, we provide a polynomial time algorithm to compute certain expected characteristic polynomials related to this construction. This leads to a deterministic polynomial time algorithm to compute bipartite Ramanujan (multi)graphs of all degrees and all sizes

    Approximating the Largest Root and Applications to Interlacing Families

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    We study the problem of approximating the largest root of a real-rooted polynomial of degree nn using its top kk coefficients and give nearly matching upper and lower bounds. We present algorithms with running time polynomial in kk that use the top kk coefficients to approximate the maximum root within a factor of n1/kn^{1/k} and 1+O(lognk)21+O(\tfrac{\log n}{k})^2 when klognk\leq \log n and k>lognk>\log n respectively. We also prove corresponding information-theoretic lower bounds of nΩ(1/k)n^{\Omega(1/k)} and 1+Ω(log2nkk)21+\Omega\left(\frac{\log \frac{2n}{k}}{k}\right)^2, and show strong lower bounds for noisy version of the problem in which one is given access to approximate coefficients. This problem has applications in the context of the method of interlacing families of polynomials, which was used for proving the existence of Ramanujan graphs of all degrees, the solution of the Kadison-Singer problem, and bounding the integrality gap of the asymmetric traveling salesman problem. All of these involve computing the maximum root of certain real-rooted polynomials for which the top few coefficients are accessible in subexponential time. Our results yield an algorithm with the running time of 2O~(n3)2^{\tilde O(\sqrt[3]n)} for all of them

    Interlacing Families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem

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    We use the method of interlacing families of polynomials introduced to prove two theorems known to imply a positive solution to the Kadison--Singer problem. The first is Weaver's conjecture KS2KS_{2} \cite{weaver}, which is known to imply Kadison--Singer via a projection paving conjecture of Akemann and Anderson. The second is a formulation due to Casazza, et al., of Anderson's original paving conjecture(s), for which we are able to compute explicit paving bounds. The proof involves an analysis of the largest roots of a family of polynomials that we call the "mixed characteristic polynomials" of a collection of matrices.Comment: This is the version that has been submitte
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