Polynomial time algorithms for some evacuation problems

Evacuation problems can be modeled as flow problems on dynamic networks. A dvnamic network is defined by a graph with capacities and integral transit times on its edges. The maximum dvnamic flow oroblem is to send a maximum amount of flow from a source to a sink within a given time bound T; conversely, the quickest flow problem is to send a given flow amount v from the source to the sink in the shortest possible time. These dynamic flow problems have been studied previously and can be solved via simple minimum cost flow computations. More complicated dynamic flow problems have numerous applications and have been studied extensively. There are no polynomial time algorithms known for many of these nroblems. includine the auickest flow problem with iust two sources, each with-a flow amount that must reacha single sink. The general multiple source quickest flow problem is commonly used as a model for building evacuation; we also call it the evacuation problem. In this paper we consider three problems related to the evacuation problem. We give a polynomial time algorithm for the evacuation problem with a fixed number of sources and sinks. We give a polynomial time algorithm for the lexicographic maximum dynamic flow problem with any number of sources: in this problem we seek a dynamic flow that lexicographically maximizes the amounts-of flow leavine the sources in a soecified order. Our algorithm for the evzcuation problem follows as an application. We also consider the earliest arrival flow problem. Given a source, sink, and time bound T, an earliest arrival flow maximizes the amount of flow reaching the sink at every time step up to and including T. The existence of such a flow is well known, but there are no polynomial time algorithms known even to approximate it. We give a polynomial time algorithm that for any fixed c > 0 approximates an earliest arrival flow within a factor of 1 +c. ‘Research was done while the authors were visiting the Department of Computer Science at Princeton University. tDepartment of Computer Science, Cornell University, Ithaca, NY 14853. Research supported by a National Science Foundation Graduate Research Fellowship. tSchoo1 of Operations Research & Industrial Engineering, Cornell University, Ithaca, NY 14853. Research supported in part by a Packard FeIIowship, an NSF PYI award, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550. Iha Tardost