4,806 research outputs found
Interlacing Families IV: Bipartite Ramanujan Graphs of All Sizes
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
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
We study the problem of approximating the largest root of a real-rooted
polynomial of degree using its top coefficients and give nearly
matching upper and lower bounds. We present algorithms with running time
polynomial in that use the top coefficients to approximate the maximum
root within a factor of and when and respectively. We also prove corresponding
information-theoretic lower bounds of and
, 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 for all
of them
Interlacing Families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem
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 \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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