Feasible Schedules for Rotating Transmissions

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. Let k, n be two integers, let wsj , 1 ≤ s ≤ n, 1 ≤ j ≤ k be non-negative reals satisfying ∑k j=1 wsj < 1/n for every 1 ≤ s ≤ n and let dsj be arbitrary non-negative reals. Then there are real numbers x1, x2, . . . , xn so that for every j, 1 ≤ j ≤ k, the n cyclic closed intervals I (j) s = [xs + dsj , xs + dsj + wsj ], (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.