Competitive Algorithms for Cottage Rental

Suppose that you are given a pool of cottages which you have for rent for a given holiday period. People make reservation requests over the phone, requesting a cottage from some time a j to b j . Given a rental request one must decide without (complete) knowledge about future requests, whether to accept this request and make a profit that is proportional to the renting duration b j − a j or to reject it. This problem is related to the seat-reservation problem (see e.g. [J. Boyar, K.S. Larsen, The seat reservation problem, Algorithmica 25 (1999) 403–417, J. Boyar, S. Krarup, M.N. Nielsen, Seat reservation allowing seat changes, Journal of Algorithms 52 (2004) 169–192]), the difference being that the specific cottage need not be fixed upon acceptance of a request. Moreover, in [J. Boyar, K.S. Larsen, The seat reservation problem, Algorithmica 25 (1999) 403–417, J. Boyar, S. Krarup, M.N. Nielsen, Seat reservation allowing seat changes, Journal of Algorithms 52 (2004) 169–192] there is a fairness restriction: it is assumed that passengers that can be accepted must be accepted. Anyone trying to reserve a cottage for one night in the middle of the high season will find out that cottage owners are not necessarily fair (in this sense). s paper we study a generalized version of the cottage rental problem. We wish to schedule a maximum profit selection of jobs on m identical machines. Jobs arrive online one by one and each job is specified by its start and end time. The goal is to determine a non-preemptive schedule which maximizes the profit of the scheduled jobs, where the profit of a job is equal to its length. Upon arrival of a new job, an online algorithm must decide whether to accept the job (“admit the job”) or not. If the job is accepted, the online algorithm must be able to reorganize its already existing schedule such that the new job can be processed together with all previously admitted jobs. The algorithm need not specify on which machine the job will eventually be run. vide lower bounds on the competitive ratio for randomized algorithms against an oblivious adversary and present deterministic algorithms which match the lower bounds up to (small) constant factors.