The Errand Scheduling Problem

We consider the following natural errand scheduling problem (ESP). We must perform a set of errands at the nodes of an edge-weighted graph, and each errand can only be performed at a subset of the nodes. What is the shortest tour that allows us to complete all the errands? We also consider the closely related tree cover problem (TCP), in which we seek a tree of minimum length that \covers" all errands. Both problems generalize a number of well-known problems in combinatorial optimization and have a wide range of applications including job scheduling on multi-state CNC machines and VLSI design. We focus on the case in which the graph is a weighted tree; this trivially generalizes the famous set cover problem. Under the assumption that no errand can be performed in \too many" nodes, we obtain an algorithm that asymptotically matches the best possible approximation ratio for set cover and approximates both errand scheduling and tree cover within O(log m), where m is the total number of errands. Our algorithm is based on \tree stripping"{a technique speciically designed to round the solution to the LP relaxation of tree cover, but is hopefully useful for approximating other related problems. In the second part of the paper, we (brieey) discuss the errand scheduling and tree cover problems on a general weighted graph with the restriction that each errand can be performed at at most nodes. We show that in this case, ESP can be approximated to within 3=2 and TCP to within .

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