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The Conditional DAG (CDAG) task model is used for modeling multiprocessor real-time systems containing conditional expressions for which outcomes are not known prior to their evaluation. Feasibility analysis for CDAG tasks upon multiprocessor platforms is shown to be complete for the complexity classpspace; assumingnp≠pspace, this result rules out the use of Integer Linear Programming solvers for solving this problem efficiently. It is further shown that there can be no pseudo-polynomial time algorithm that solves this problem unlessp=pspace.

We introduce and study a class of optimization problems we call replenishment problems with fixed turnover times: a very natural model that has received little attention in the literature. Clients with capacity for storing a certain commodity are located at various places; at each client the commodity depletes within a certain time, the turnover time, which is constant but can vary between locations. Clients should never run empty. The natural feature that makes this problem interesting is that we may schedule a replenishment (well) before a client becomes empty, but then the next replenishment will be due earlier also. This added workload needs to be balanced against the cost of routing vehicles to do the replenishments. In this paper, we focus on the aspect of minimizing routing costs. However, the framework of recurring tasks, in which the next job of a task must be done within a fixed amount of time after the previous one is much more general and gives an adequate model for many practical situations. Note that our problem has an infinite time horizon. However, it can be fully characterized by a compact input, containing only the location of each client and a turnover time. This makes determining its computational complexity highly challenging and indeed it remains essentially unresolved. We study the problem for two objectives:min–avg  minimizes the average tour cost andmin–max  minimizes the maximum tour cost over all days. Formin–max  we derive a logarithmic factor approximation for the problem on general metrics and a 6-approximation for the problem on trees, for which we have a proof of NP-hardness. Formin–avg  we present a logarithmic factor approximation on general metrics, a 2-approximation for trees, and a pseudopolynomial time algorithm for the line. Many intriguing problems remain open.