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 $n$ using its top $k$ coefficients and give nearly matching upper and lower bounds. We present algorithms with running time polynomial in $k$ that use the top $k$ coefficients to approximate the maximum root within a factor of $n^{1/k}$ and $1+O(\tfrac{\log n}{k})^2$ when $k\leq \log n$ and $k>\log n$ respectively. We also prove corresponding information-theoretic lower bounds of $n^{\Omega(1/k)}$ and $1+\Omega\left(\frac{\log \frac{2n}{k}}{k}\right)^2$, 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 $2^{\tilde O(\sqrt[3]n)}$ for all of them.