How to Fit In Another Meeting

We are studying the problem of determining suitable meeting times and locations for a group of participants wishing to schedule a new meeting subject to already scheduled meetings possibly held at a number of different locations. Each participant must be able to reach the new meeting location, attend for the entire duration, and reach the next meeting location on time. In particular, we give a solution to the problem instance where each participant has two scheduled meetings separated by a free time interval. For a geometric model, where n participants can travel along straight paths in the Euclidean plane, we present an O(n log n) algorithm to determine the longest meeting duration and a location suitable to all participants. In a graph-based model, transportation is provided by a geometric network over m nodes and e edges in the plane. Participants can have individual weights. Moreover, there can be k groups of participants, such that only one member of each group must attend the meeting. In this model, a location for a meeting of longest possible duration can be determined in time O(en¿(k) log k + n log n + mn log m), where a (k) denotes the extremely slowly growing inverse Ackermann function.