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This paper studies the aggregation of messages in networks that consist of a chain of nodes, and each message is time-constrained such that it needs to be aggregated during a given time interval, called its due interval. The objective is to minimize the maximum cost incurred at any node, which is for example a concern in wireless sensor networks, where it is crucial to distribute the energy consumption as equally as possible. First, we settle the complexity of this problem by proving its NP-hardness, even for the case of unit length due intervals. Second, we give a QPTAS, which we extend to a PTAS for the special case that the lengths of the due intervals are constants. This is in particular interesting, since we prove that this problem becomes APX-hard if we consider tree networks instead of chain networks, even for the case of unit length due intervals. Specifically, we show that it cannot be approximated within 4/3 *** *** for any *** 0 , unless P=NP.