A Note on the Complexity of Scheduling for Precedence Constrained Messages in Distributed Systems

This note considers a problem of minimum length scheduling for a set of messages subject to precedence constraints for switching and communication networks. The problem was first studied by Barcaccia, Bonuccelli, and Di Iannii [1]. We consider a network with n inputs and n outputs. The messages to be sent are represented by an n × n matrix D = [di j], the traffic matrix, whose entries are nonnegative integers, where 0 ≤ i, j ≤ n − 1. Entry di j represents the number of messages to be sent from input i to output j. In order to specify precedence constraints among messages, we represent a traffic matrix D by a sequence of n× n matrices D = (D(0), D(1), . . . , D(k−1)) such that D = ∑k−1 r=0 D (r). We consider precedence constraints on the rows, which means that the entries in each row of D(r+1) can be scheduled only if the entries in the corresponding row of D(r) have already been scheduled (0 ≤ r ≤ k − 2). A switching matrix is a binary matrix with at most one nonzero entry in each row and in each column. A switching matrix represents messages that can be sent simultaneously without conflicts. A sequence of n × n switching matrices S = (S (0), S (1), . . . , S (t−1)) is called a switching schedule for D if the following conditions are satisfied: