A 2.542-approximation for precedence constrained single machine scheduling with release dates and total weighted completion time objective

Abstract

We present a $\sqrt{e}/(\sqrt{e}-1)$-approximation algorithm for the nonpreemptive scheduling problem to minimize the total weighted completion time of jobs on a single machine subject to release dates and precedence constraints. The previously best known approximation algorithm dates back to 1997; its performance guarantee can be made arbitrarily close to the Euler constant  $e$  (Schulz and Skutella, 1997).

Introduction

We consider the following classical machine scheduling problem denoted by $1|r_j,\mathrm{prec}|\sum w_j{C}_j$ in the standard classification scheme of Graham, Lawler, Lenstra, and Rinnooy Kan  [12]. We are given a set of jobs $N=\{1,2,\dots ,n\}$ and for every job $j\in N$ a processing time $p_j\ge 0$, a release date $r_j\ge 0$, and a weight $w_j\ge 0$. The jobs $j\in N$ need to be processed during non-overlapping time intervals of length $p_j$, and $j$’s processing must not start before its release date $r_j$. Moreover, there are precedence constraints given by a partial order “$\prec$” on $N$ where $j\prec k$ means that job $j$ must be completed before job $k$ may be started, that is, $j$’s processing interval must precede $k$’s. We may therefore without loss of generality assume throughout the paper that $j\prec k$ implies $r_j\le r_k$. The objective is to minimize the total weighted completion time ${\sum }_{j\in N}{w}_j{C}_j$ where $C_j$ denotes the first point in time at which $j$’s processing is completed.

Complexity. Even for unit job weights, the special cases of the problem without non-trivial release dates $1\left|\mathrm{prec}\right|\sum C_j$ (i.e., $r_j=0$ for all $j\in N$) or without precedence constraints $1\left|r_j\right|\sum C_j$ are strongly NP-hard; see, e.g.,  [8, problem SS4]. In preemptive scheduling, the processing of a job may be repeatedly interrupted and resumed at a later point in time. In the absence of precedence constraints, the problem with unit job weights $1|r_j,\mathrm{pmtn}|\sum C_j$ can be solved in polynomial time  [3], but for arbitrary weights $1|r_j,\mathrm{pmtn}|\sum w_j{C}_j$ is strongly NP-hard. Without non-trivial release dates preemptions are superfluous such that $1|\mathrm{prec},\mathrm{pmtn}|\sum C_j$ is equivalent to $1\left|\mathrm{prec}\right|\sum C_j$ and thus strongly NP-hard.

List scheduling. Before dipping into the rich history of approximation algorithms for these scheduling problems, we first discuss the most important algorithmic ingredient for both heuristic and exact solutions: list scheduling. Consider a list representing a total order on the set of jobs $N$, extending the given partial order “$\prec$”. A straightforward way to construct a feasible schedule is to process the jobs in the given order as early as possible with respect to release dates. A schedule constructed in this way is a list schedule.

Depending on the given list and the release dates of jobs, the machine might remain idle when one job is completed but the next job in the list is not yet released. On the other hand, if job preemptions are allowed, it is certainly not advisable to leave the machine idle while another job at a later position in the list is already available (released) and waiting. Instead, we better start this job and preempt it from the machine as soon as the next job in the list is released. In preemptive list scheduling we process at any point in time the first available job in the list. The resulting preemptive schedule is feasible (as $j\prec k$ implies $r_j\le r_k$) and is called preemptive list schedule.

Known techniques and results. There is a vast literature on approximation algorithms for the various scheduling problems mentioned above. Here we only mention those results that are particularly relevant in the context of this paper and refer to Chekuri and Khanna  [5] for a more comprehensive overview. Various kinds of linear programming (LP) relaxations have proved to be useful in designing approximation algorithms. One of the simplest and most intuitive classes of LP relaxations is based on completion time variables only. These LP relaxations were introduced by Queyranne  [16] and first used in the context of approximation algorithms by Schulz  [17], who presents a 2-approximation algorithm for the problem $1\left|\mathrm{prec}\right|\sum w_j{C}_j$ and a 3-approximation algorithm for $1|r_j,\mathrm{prec}|\sum w_j{C}_j$; see also Hall, Schulz, Shmoys, and Wein  [13]. These algorithms compute an optimal LP solution and then do list scheduling in order of increasing LP completion times. Moreover, Hall et al.  [13] show that preemptive list scheduling in order of increasing LP completion times is a 2-approximation algorithm for $1|r_j,\mathrm{prec},\mathrm{pmtn}|\sum w_j{C}_j$.

Phillips, Stein, and Wein  [15] and Hall, Shmoys, and Wein  [14] introduce the idea of list scheduling in order of so-called $\alpha$-points to convert preemptive schedules to nonpreemptive ones. For $\alpha \in (0,1]$, the $\alpha$-point of a job with respect to a preemptive schedule is the first point in time when an $\alpha$-fraction of the job has been completed. Goemans  [10] and Chekuri, Motwani, Natarajan, and Stein  [6] show that choosing $\alpha$ randomly leads to better results. In particular, Chekuri et al.  [6] present an $e/(e-1)$-approximation algorithm for $1\left|r_j\right|\sum C_j$ by starting from an optimal preemptive schedule. Goemans  [10] and Goemans, Queyranne, Schulz, Skutella, and Wang  [11] give approximation results for the more general weighted problem $1\left|r_j\right|\sum w_j{C}_j$ based on a preemptive schedule that is an optimal solution to an LP relaxation in time-indexed variables. Similarly, Schulz and Skutella  [18] give an $(e+\epsilon )$-approximation algorithm for $1|r_j,\mathrm{prec}|\sum w_j{C}_j$ for any $\epsilon >0$.

Bansal and Khot prove in a recent landmark paper  [4] that there is no $(2-\epsilon )$-approximation algorithm for $1\left|\mathrm{prec}\right|\sum w_j{C}_j$, assuming a stronger version of the Unique Games Conjecture. Ambühl, Mastrolilli, Mutsanas, and Svensson  [2], based on earlier work of Correa and Schulz  [7] and Ambühl and Mastrolilli  [1], prove an interesting relation between the approximability of $1\left|\mathrm{prec}\right|\sum w_j{C}_j$ and the vertex cover problem

Our contribution. We present a $\sqrt{e}/(\sqrt{e}-1)$-approximation algorithm for the problem $1|r_j,\mathrm{prec}|\sum w_j{C}_j$ based on the following two ingredients: (i) For the problem $1|r_j,\mathrm{prec},\mathrm{pmtn}|\sum w_j{C}_j$ we slightly strengthen the 2-approximation result of Hall et al.  [13] and show that preemptive list scheduling in order of increasing LP completion times on a machine running at double speed yields a schedule whose cost is at most the cost of an optimal schedule on a regular machine; see Section  2. (ii) Modifying the analysis of Chekuri et al.  [6] we show how to turn the preemptive schedule on the double speed machine into a nonpreemptive schedule on a regular machine while increasing the objective function by at most a factor of $\sqrt{e}/(\sqrt{e}-1)$; see Section  3. We conclude with a conjecture in Section  4.