Online Linear Optimization over Permutations with Precedence Constraints

We consider an online linear optimization problem over the set of permutations under some precedence constraints. In this problem, the player is supposed to predict a permutation of n fixed objects at each trial, under the constraints that some objects have higher priority than other objects in each permutation. This problem is naturally motivated by a scheduling problem whose objective is to minimize the sum of completion times of n sequential tasks under precedence constraints. We propose an online linear optimization algorithm which predicts almost as well as the best known offline approximation algorithms in hindsight. Furthermore, our algorithm runs in O(n) time for each trial.