TY - JOUR

T1 - Using fractional primal-dual to schedule split intervals with demands

AU - Bar-Yehuda, Reuven

AU - Rawitz, Dror

PY - 2006/12/1

Y1 - 2006/12/1

N2 - 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 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.

AB - 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 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.

KW - Approximation algorithms

KW - Local ratio

KW - Primal-dual

KW - Scheduling

KW - t-intervals

UR - http://www.scopus.com/inward/record.url?scp=33748452123&partnerID=8YFLogxK

U2 - 10.1016/j.disopt.2006.05.010

DO - 10.1016/j.disopt.2006.05.010

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AN - SCOPUS:33748452123

SN - 1572-5286

VL - 3

SP - 275

EP - 287

JO - Discrete Optimization

JF - Discrete Optimization

IS - 4

ER -