Finding the Smallest Gap between Sums of Square Roots

Abstract

Let k and n be positive integers, n > k. Define r(n,k) to be the minimum positive value of

$$ |\sqrt{a_1} + \cdots + \sqrt{a_k} - \sqrt{b_1} - \cdots -\sqrt{b_k} | $$

where a ₁, a ₂, ⋯ , a _k , b ₁, b ₂, ⋯ , b _k are positive integers no larger than n. It is important to find a tight bound for r(n,k), in connection to the sum-of-square-roots problem, a famous open problem in computational geometry. The current best lower bound and upper bound are far apart. In this paper, we present an algorithm to find r(n,k) exactly in n ^k + o(k) time and in n ^{⌈k/2 ⌉ + o(k)} space. As an example, we are able to compute r(100,7) exactly in a few hours on one PC. The numerical data indicate that the known upper bound seems closer to the truth value of r(n,k).

This research is partially supported by NSF grant CCF-0830522 and CCF-0830524.