Using Fractional Primal-Dual to Schedule Split Intervals with Demands

Given a limited resource (such as bandwidth or memory) we consider the problem of scheduling requests from clients that are given as groups of non-intersecting time intervals. Each request j is associated with a demand (the amount of resource required), dj , a t-interval , which consists of up to t segments, for some t ≥ 1, and a weight, w(j). A schedule is a collection of requests. It is feasible if for every time instance s, the total demand of scheduled requests whose t-interval contains s does not exceed 1, the amount of resource available. Our goal is to find a feasible schedule that maximizes the total weight of scheduled requests. This problem generalizes many problems from the literature, and show up in a wide range of applications. We present a 6t-approximation algorithm for this problem that uses a novel extension of the primal-dual schema we call fractional primal-dual . A fractional primal-dual algorithm produces a primal solution x, and a dual solution y, whose value divided by r, the approximation ratio, bounds the weight of x. However, y is not a solution of the dual of an LP relaxation of the problem. The algorithm induces a new LP that has the same objective function as the original problem, but contains inequalities that may not be valid with respect to the original problem. y is a feasible solution of the dual of the new LP. x is r-approximate, since some optimal solution of an LP relaxation of the original problem is in the feasible set of this new LP. We present a fractional local ratio interpretation of our 6t-approximation algorithm. We also discuss the connection between fractional primal-dual and the fractional local ratio technique. Specifically, we show that the former is the primal-dual manifestation of latter. Department of Computer Science, Technion, Haifa 32000, Israel. E-mail: [email protected]. Department of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel. E-mail: [email protected].