Single Machine Scheduling with Release Dates

Abstract

We consider the scheduling problem of minimizing the average weighted completion time of n jobs with release dates on a single machine. We first study two linear programming relaxations of the problem, one based on a time-indexed formulation, the other on a completiontime formulation. We show their equivalence by proving that a O(n log n) greedy algorithm leads to optimal solutions to both relaxations. The proof relies on the notion of mean busy times of jobs, a concept which enhances our understanding of these LP relaxations. Based on the greedy solution, we describe two simple randomized approximation algorithms, which are guaranteed to deliver feasible schedules with expected objective value within factors of 1.7451 and 1.6853, respectively, of the optimum. They are based on the concept of common and independent a-points, respectively. The analysis implies in particular that the worst-case relative error of the LP relaxations is at most 1.6853, and we provide instances showing that it is at least e/(e-1) 1.5819. Both algorithms may be derandomized, their deterministic versions running in O(n 2) time. The randomized algorithms also apply to the on-line setting, in which jobs arrive dynamically over time and one must decide which job to process without knowledge of jobs that will be released afterwards.

LP-Based Online Scheduling: From Single to Parallel Machines

“…Using the idea of α-points and mean-busy-time relaxations, Goemans et al [11] designed a deterministic 2.4143-competitive and a randomized 1.6853-competitive algorithm for the online 1|r j | j w j C j . A similar approach was taken by Schulz and Skutella [22] to give a randomized 4 3 -competitive algorithm for 1|r j , pmtn| j w j C j ; Sitters [24] gave a 1.56-competitive deterministic algorithm for the same problem.…”

Due Date Management Policies

“…An off-line algorithm A is called a ρ − approximation, if it delivers a solution of quality at most ρ times optimal (for minimization problems). Although the use of competitive analysis in studying DDM problems have been limited, it has received much attention recently in the scheduling literature [43] [44] [53].…”

Combinatorial algorithms for minimizing the weighted sum of completion times on a single machine

“…For the offline problem, polynomial time approximation schemes have been developed [1]. There are also several linear programming based approximations [3,4,7,10,18,19,20]. The linear programming based techniques derive primarily from three different LP formulations (discussed in detail in [17]): the completion time LP (LP1), the completion time LP with shifted parallel inequalities (LP2), and the preemptive time indexed LP (LP3).…”

“…The linear programming based techniques derive primarily from three different LP formulations (discussed in detail in [17]): the completion time LP (LP1), the completion time LP with shifted parallel inequalities (LP2), and the preemptive time indexed LP (LP3). Most algorithms use LP2 and LP3 [3,4,7,19,20]. Goemans et al [7] study these two LP formulations in detail, show their equivalence and present a LP-rounding algorithm with an approximation guarantee of 1.6853.…”

“…Most algorithms use LP2 and LP3 [3,4,7,19,20]. Goemans et al [7] study these two LP formulations in detail, show their equivalence and present a LP-rounding algorithm with an approximation guarantee of 1.6853. They also give an online algorithm with a competitive ratio of (1 + √ 2).…”