view of our impossibility result it might also be of interest to study the impact of the amount of locality information in the static case, and in the dynamic case, in which locality refers also to the dimension of time, that is, the amount of local information regarding the occurrence of the messages. 26 follow only one path between them (as is the case in certain situations, like in certain ATM routing problems), then the resulting problem is NP-complete, even when the network is bottleneck-free, and the processors have global information about the initial connguration. Theorem 15 Given a network G = (V; E), given capacities c(e) for every edge e 2 E, and given integers a i;j 0, for i; j 2 V , it is NP-Complete to determine whether there exists a feasible schedule in which, for every i; j 2 V , a i;j messages are sent from i to j, all following the same path between i and j, such that each meets its given deadline and such that no more than c(e) messages are delivered along any edge e in any unit of time. Proof: We use a reduction from the Partition problem (see 5]): Given a set of integers fa i ji = 1; ; 2ng, is it possible to nd a subset I f1; ; 2ng such that P i2I a i = P i= 2I a i ? Given an instance of this problem, we deene a network, in which processor a is connected to processor b by two parallel lines e 1 and e 2 , and b is connected to 2n processors p 1 through p 2n. It is required to send a i messages from a to p i , for i = 1; ; 2n. The capacity of each link is 1 2 P 1i2n a i , and the deadline for each message is 1. It is possible to deliver all messages within one time unit if and only if some messages can be sent on link e 1 and the rest of them on link e 2 at time 0, and then at time 1 each message is sent by b to its destination, thus meeting its deadline. Due to the capacities of the links e 1 and e 2 , this can happen if and only if there is a subset I f1; ; 2ng …
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