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 1992 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 T¹, T²,…, T^k of tasks is a sequence s₁, s₂,…,s_k of states where s_i is the state in which Tⁱ 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 T¹T²…Tⁱ. 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|).