ESA 2005: Algorithms – ESA 2005 pp 714-725

Using Fractional Primal-Dual to Schedule Split Intervals with Demands

• Reuven Bar-Yehuda
• Dror Rawitz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3669)

Abstract

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, dj ∈ [0,1], and a weight, w(j). A schedule is a collection of jobs, such that, for every $$s \in {\mathbb R}$$, the total demand of the jobs in the schedule whose t-interval contains s does not exceed 1. Our goal is to find a schedule that maximizes the total weight of scheduled jobs.

We present a 6t-approximation algorithm 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′. 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 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 the latter.

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