On the hardness of approximating optimum schedule problems in store and forward networks

Problems related to message communication and traffic control have been assuming more and more importance due the massive use of computer networks. Scheduling a set of messages in a store and forward network means assigning to them network resources in order to deliver each message to its respective destination. The goal of typical scheduling problems is to devise strategies of assignments that minimize the delivery time of all messages or, alternatively, their total end-to-end delay. Both problems have been proven to be network performance (NP)-hard even under very restrictive hypothesis. We study the computational complexity of approximating the minimum end-to-end delay. Unfortunately, it turns out that finding an approximated solution with approximation ratio smaller than k^1/10 is as difficult as finding the optimal solution (where k is the number of messages in the network). More precisely, approximating the optimum delay is NP-hard even when a nonconstant approximation ratio is allowed. This result holds also in the case of layered networks. We then prove that if we consider a particular class of local schedules (i.e., distributed strategies that can only use information about limited regions of the network) then the approximation error cannot be bounded by any sublinear function in k. Such results also provide a lower bound on the approximation of the minimum delivery time problem. Thus, if the attention is restricted to polynomial-time algorithms, the only possibility is designing heuristics that behave well on average. As a first step to this aim, we evaluate the expected approximation error of some simple heuristics using several experimental tests