68-06 Proceedings, conferences, collections, etc.
We study the fundamental problem of scheduling bidirectional traffic across machines arranged on a path. The main feature of the problem is that jobs traveling in the same direction can be scheduled in quick succession on a machine, while jobs in the other direction have to wait for an additional transit time. We show that this tradeoff makes the problem significantly harder than the related flow shop problem, by showing that it is NP-hard even for jobs with identical processing and transit times. We give polynomial algorithms for a single machine and any constant number of machines. In contrast, we show the problem to be NP-hard on a single machine and with identical processing and transit times if some pairs of jobs in different directions are allowed to run on the machine concurrently. We generalize a PTAS of Afrati et al. [1999] for one direction and a single machine to the bidirectional case on any constant number of machines.
We propose a new approach to competitive analysis by introducing the novel concept of online approximation schemes. Such scheme algorithmically constructs an online algorithm with a competitive ratio arbitrarily close to the best possible competitive ratio for any online algorithm. We study the problem of scheduling jobs online to minimize the weighted sum of completion times on parallel, related, and unrelated machines, and we derive both deterministic and randomized algorithms which are almost best possible among all online algorithms of the respective settings. Our method relies on an abstract characterization of online algorithms combined with various simplifications and transformations. We also contribute algorithmic means to compute the actual value of the best possible competitive ratio up to an arbitrary accuracy. This strongly contrasts all previous manually obtained competitiveness results for algorithms and, most importantly, it reduces the search for the optimal competitive ratio to a question that a computer can answer. We believe that our method can also be applied to many other problems and yields a completely new and interesting view on online algorithms.