An optimal on-line algorithm for metrical task system

In practice, almost all dynamic systems require decisions to be made on-line, without full knowledge of their future impact on the system. A general model for the processing of sequences of tasks is introduced, and a general on-line decision algorithm is developed. It is shown that, for an important class of special cases, this algorithm is optimal among all on-line algorithms. Specifically, a task system ( S,d ) for processing sequences of tasks consists of a set S of states and a cost matrix d where d ( i, j is the cost of changing from state i to state j (we assume that d satisfies the triangle inequality and all diagonal entries are 0). The cost of processing a given task depends on the state of the system. A schedule for a sequence T 1 , T 2 ,…, T k of tasks is a sequence s 1 , s 2 ,…, s k of states where s i is the state in which T i is processed; the cost of a schedule is the sum of all task processing costs and the state transition costs incurred. An on-line scheduling algorithm is one that chooses s i only knowing T 1 T 2 … T i . Such an algorithm is w -competitive if, on any input task sequence, its cost is within an additive constant of w times the optimal offline schedule cost. The competitive ratio w ( S , d ) is the infimum w for which there is a w -competitive on-line scheduling algorithm for ( S , d ). It is shown that w ( S , d ) = 2|S|–1 for every task system in which d is symmetric, and w ( S, d ) = O (| S | 2 ) for every task system. Finally, randomized on-line scheduling algorithms are introduced. It is shown that for the uniform task system (in which d ( i,j ) = 1 for all i,j ), the expected competitive ratio w¯ ( S,d ) = O (log|S|).