# Approximating the Largest Root and Applications to Interlacing Families

## TL;DR

This paper develops algorithms to approximate the largest root of real-rooted polynomials using only top coefficients, with applications to interlacing families, Ramanujan graphs, Kadison-Singer, and TSP, providing near-optimal bounds.

## Contribution

It introduces nearly tight bounds and efficient algorithms for approximating the largest root from limited polynomial coefficients, impacting several key areas in combinatorics and optimization.

## Key findings

- Algorithms approximate the maximum root within specific factors depending on k.
- Matching lower bounds establish the near-optimality of the algorithms.
- Applications include improved algorithms for problems related to interlacing families and graph theory.

## Abstract

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.