Motivated by a scheduling problem that arises in the study of optical networks, we prove the following result, which is a variation of a conjecture of Haxell, Wilfong and Winkler. et k, n be two positive integers, let w sj, 1 ≤ s ≤ n, 1 ≤ j ≤ k be nonnegative reals satisfying Σj=1k wsj < 1/n every 1 ≤ s ≤ n and let dsj be arbitrary nonnegative reals. Then there are real numbers x1, x2,...,xn such that for every j, 1 ≤j ≤ k, the n cyclic closed intervals Is(j) = [xs + dsj, xs + ddj + w sj], (1 ≤ s ≤ n), where the endpoints are reduced modulo 1, are pairwise disjoint on the unit circle. The proof is based on some properties of multivariate polynomials and on the validity of the Dyson conjecture.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics