Discrete Optimization
Two simple constant ratio approximation algorithms for minimizing the total weighted completion time on a single machine with a fixed non-availability interval

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Abstract

In this note, we consider the scheduling problem of minimizing the sum of the weighted completion times on a single machine with one non-availability interval on the machine under the non-resumable scenario. Together with a recent 2-approximation algorithm designed by Kacem [I. Kacem, Approximation algorithm for the weighted flow-time minimization on a single machine with a fixed non-availability interval, Computers & Industrial Engineering 54 (2008) 401–410], this paper is the first successful attempt to develop a constant ratio approximation algorithm for this problem. We present two approaches to designing such an algorithm. Our best algorithm guarantees a worst-case performance ratio of 2+ε.

Introduction

In this paper, we address the problem of minimizing the sum of the weighted completion times on a single machine, provided that the machine is not available for processing during a given time interval and the job that cannot complete before that interval restarts from scratch when the machine becomes available again.

Our scheduling model belongs to the family of scheduling models with machine availability constraints. We refer to a recent survey by Lee [12] for a state-of-the-art review of deterministic scheduling under availability constraints. Assume that for the processing machine there is a known non-availability interval [s, t], during which the machine cannot perform the processing of any job. This non-availability interval can be due to some scheduled activity other than processing (a maintenance period, a rest period, etc.). The job that is affected by the non-availability interval is called the crossover job. There are several possible scenarios to handle the crossover job. Under the non-resumable scenario, the crossover job that cannot be completed by time s is restarted from scratch at time t. Under the resumable scenario the crossover job is interrupted at time s and resumed from the point of interruption at time t.

As an illustration of the single machine problem with a single non-availability interval under the non-resumable scenario, consider the following situation. A student uses her laptop to download files from an Internet site. She will need her laptop from 1 pm to 2 pm to do a presentation of her project, so that the laptop cannot be used for downloading during this time slot. In the case that she fails to completely download a file by 1 pm, she will start downloading that file again after 2 pm.

Formally, we are given a set N={1,2,,n} of jobs that have to be processed on a single machine. The processing of job jN takes pj time units. There is a weight wj associated with job j, which indicates its relative importance. All values pj and wj are positive integers. The machine processes at most one job at a time. The completion time of job jN in a feasible schedule S is denoted by Cj(S), or shortly Cj if it is clear which schedule is referred to. It is required to minimize the total weighted completion time, i.e., the functionZ(S)=j=1nwjCj(S).Extending standard scheduling notation, we denote the resulting problem under the non-resumable scenario by 1|h(1),N-res|wjCj, and that under the resumable scenario by 1|h(1),Res|wjCj. A schedule S is optimal if Z(S)Z(S) for any feasible schedule S. Both problems are NP-hard in the ordinary sense, see [1], [11].

The fact that the problems under consideration are NP-hard calls for design and analysis of approximation algorithms. Recall that for a problem of minimizing a function Z(x), where x is a collection of decision variables, a polynomial time algorithm that finds a solution xH such that Z(xH) is at most ρ1 times the optimal value Z(x) is called a ρ-approximation algorithm; the value of ρ is called a worst-case ratio bound. Given an ε>0, a family of (1+ε)-algorithms is called a fully polynomial time approximation scheme (FPTAS) if its running time depends polynomially on the size of the input of the problem and 1/ε.

For problem 1|h(1),N-res|wjCj with the machine non-availability interval I=[s,t], assume that the jobs are numbered in such a way thatp1w1p2w2pnwn.We call the sequence of jobs numbered in accordance with (1) a Smith sequence or a WSPT sequence (weighted shortest processing time). It is well-known that in an optimal schedule for the classical problem 1wjCj of minimizing the sum of the weighted completion times on a single permanently available machine the jobs are processed according to the WSPT sequence, see, e.g. [15].

Lee [11] proves that for problem 1|h(1),Res|wjCj there exists an optimal schedule in which the jobs sequenced before interval I and after that interval obey the WSPT rule. Furthermore, if the jobs are scheduled according to their numbering (1) and the crossover job restarts from scratch at time t, then for the resulting schedule SWSPT the ratio Z(SWSPT)/Z(S) can be arbitrarily large; see [11]. For problem 1|h(1),N-res|wjCj, effective branch and bound algorithms are developed in [5], [6]. Kacem and Chu [4] study the WSPT rule and its modification and give conditions under which these heuristics behave as 3-approximation algorithms. Together with a recent 2-approximation algorithm designed by Kacem [3], this paper is the first successful attempt to develop a constant ratio approximation algorithm for the general problem 1|h(1),N-res|wjCj.

For the unweighted case, the numbering (1) is equivalent to the SPT (shortest processing time) rule. Lee and Liman [13] by refining the analysis by Adiri et al. [1] demonstrate that for problem 1|h(1),N-res|Cj the schedule found by the SPT rule guarantees that Z(SSPT)/Z(S)97, and this bound is tight. For problem 1|h(1),N-res|Cj, an algorithm with a worst-case performance ratio of 2017 is given by Sadfi et al. [14], and that has been recently further improved by Breit [2].

Problem 1|h(1),Res|wjCj in which the crossover job is dealt with according to the resumable scenario is studied by Lee [11]. He shows that this problem is NP-hard, but its counterpart with equal weights, i.e., problem 1|h(1),Res|Cj, is solvable in O(nlogn) time, since schedule SSPT is optimal. Similarly to the non-resumable scenario, for problem 1|h(1),Res|wjCj there exists an optimal schedule in which the jobs sequenced before and after the crossover job follow the WSPT rule. For problem 1|h(1),Res|wjCj, Wang et al. [16] have recently reported a 2-approximation algorithm that requires O(n2) time.

The remainder of this paper is organized as follows. In Section 2, we show that a ρ-approximation algorithm for problem 1|h(1),Res|wjCj can be converted into a 2ρ-approximation algorithm for problem 1|h(1),N-res|wjCj. In particular, this gives a 4-approximation algorithm for the latter problem. On the other hand, we demonstrate that the worst-case performance ratio delivered by any algorithm of this type cannot be smaller than 2. In Section 3, we present a simple (2+ε)-approximation algorithm for problem 1|h(1),N-res|wjCj based on an approximate solution of a linear Knapsack problem. Section 4 contains some concluding remarks.

Section snippets

A heuristic based on the resumable scenario

In this section, we study a possibility of converting a schedule that is feasible for the resumable scenario into a schedule that is feasible under the non-resumable scenario.

For problem 1|h(1),N-res|wjCj with the non-resumable scenario we introduce the associated problem 1|h(1),Res|wjCj with the resumable scenario. These two problems share the same data set, i.e., the values pj and wj for jN, and in both problems the same non-availability interval I=[s,t] is given. Let SN denote an optimal

A heuristic based on the Knapsack problem

In this section, we present an algorithm for problem 1|h(1),N-res|wjCj that for any positive ε behaves as a (2+ε)-approximation algorithm.

Given problem 1|h(1),N-res|wjCj with the non-availability interval I=[s,t], for each job j, 1jn, introduce a Boolean variable xj in such a way thatxj=1,if jobjcompletes before intervalI,0,otherwiseRecall that we may only concentrate on schedules in which the jobs before interval I and those after interval I are sequenced according to the WSPT rule, i.e.,

Conclusion

In this note, we present approximation algorithms for the scheduling problem 1|h(1),N-res|wjCj of minimizing the sum of the weighted completion times on a single machine with one non-availability interval and the non-resumable scenario. Our purpose was to demonstrate the existence of constant ratio approximation algorithm for the problem rather than to design a practical algorithm that would deliver a small worst-case ratio. Our best algorithm is a (2+ε)-approximation that runs in O(nlogn)

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