Using fractional primal-dual to schedule split intervals with demands

We consider the problem of scheduling jobs that are given as groups of non-intersecting intervals on the real line. Each job j is associated with a t-interval (which consists of up to t segments, for some t ≥ 1), a demand, d_j ∈ [0, 1], and a weight, w (j). A feasible schedule is a collection of jobs such that, for every s ∈ R, the total demand of the jobs in the schedule whose t-interval contains s does not exceed 1. Our goal is to find a feasible schedule that maximizes the total weight of scheduled jobs. We present a 6 t-approximation algorithm for this problem that uses a novel extension of the primal-dual schema called fractional primal-dual. The first step in a fractional primal-dual r-approximation algorithm is to compute an optimal solution, x^*, of an LP relaxation of the problem. Next, the algorithm produces an integral primal solution x, and a new LP, denoted by P′, that has the same objective function as the original problem, but contains inequalities that may not be valid with respect to the original problem. Moreover, x^* is a feasible solution of P′. The algorithm also computes a solution y to the dual of P′. The solution x is r-approximate, since its weight is bounded by the value of y divided by r. We present a fractional local ratio interpretation of our 6 t-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 the latter.