On the complexity of adjacent resource scheduling

We study the problem of scheduling resource(s) for jobs in an adjacent manner (ARS). The problem relates to fixed-interval scheduling on one hand, and to the problem of two-dimensional strip packing on the other. Further, there is a close relation with multiprocessor scheduling. A distinguishing characteristic is the constraint of resource-adjacency. As an application of ARS, we consider an airport where passengers check in for their flight, joining lines before one or more desks, at the desk the luggage is checked and so forth. To smoothen these operations the airport maintains a clear order in the waiting lines: a number n(f) of adjacent desks is to be assigned exclusively during a fixed time-interval I(f) to flight f. For each flight in a given planning horizon of discrete time periods, one seeks a feasible assignment to adjacent desks and the objective is to minimize the total number of involved desks. The paper explores two problem variants and relates them to other scheduling problems. The basic, rectangular version of ARS is a special case of multiprocessor scheduling. The other problem is more general and it does not fit into any existing scheduling model. After presenting an integer linear program for ARS, we discuss the complexity of both problems, as well as of special cases. The decision version of the rectangular problem remains strongly NP-complete. The complexity of the other problem is already strongly NP-complete for two time periods. The paper also determines a number of cases that are solvable in polynomial time.