Scheduling to Minimize Total Weighted Completion Time: Performance Guarantees of L P - B a s e d Heuristics and Lower Bounds Andreas S. Schulz Technische Universit~it Berlin, Fachbereich Mathematik (MA 6-1), Strafie des 17. Juni 136, D-10623 Berlin, Germany,
[email protected]
A b s t r a c t . There has been recent success in using polyhedral formulations of scheduling problems not only to obtain good lower bounds in practice but also to develop provably good approximation algorithms. Most of these formulations rely on binary decision variables that are a kind of assignment variables. We present quite simple polynomialtime approximation algorithms that are based on linear programming formulations with completion time variables and give the best known performance guarantees for minimizing the total weighted completion time in several scheduling environments. This amplifies the importance of (appropriate) polyhedral formulations in the design of approximation algorithms with good worst-case performance guarantees. In particular, for the problem of minimizing the total weighted completion time on a single machine subject to precedence constraints we present a polynomial-time approximation algorithm with performance ratio better than 2. This outperforms a (4 + e)-approximation algorithm very recently proposed by Hall, Shmoys, and Wein that is based on time-indexed formulations. A slightly extended formulation leads to a performance guarantee of 3 for the same problem but with release dates. This improves a factor of 5.83 for the same problem and even the 4approximation algorithm for the problem with release dates but without precedence constraints, both also due to Hall, Shmoys, and Wein. By introducing new linear inequalities, we also show how to extend our technique to parallel machine problems. This leads, for instance, to the best known approximation algorithm for scheduling jobs with release dates on identical parallel machines. Finally, for the flow shop problem to minimize the total weighted completion time with both precedence constraints and release dates we present the first approximation algorithm that achieves a worst-case performance guarantee that is linear in the number of machines. We even extend this to multiprocessor flow shop scheduling. The proofs of these results also imply guarantees for the lower bounds obtained by solving the proposed linear programming relaxations. This emphasizes the strength of linear programming formulations using completion time variables.
302 1
Introduction
An important goal in real-world scheduling is to optimize "average" performance. One of the classic criteria is the average weighted completion time, or equivalently, the total weighted completion time. Approximation algorithms for scheduling problems with this objective have very recently achieved much attention, see [PSW95, HSW96]. In this paper, we present new p-approximation algorithms for a variety of NP-hard scheduling problems to minimize the total weighted completion time. That is, these algorithms have a polynomial running time and produce schedules whose objective function value is not worse than p times the optimal one. Here, p is a constant in case of a single machine or identical parallel machines, and a subtinear or a linear function in the number of machines in case of uniform parallel machines or a (muttiprocessor) flow shop environment, respectively. In particular, Hall, Shmoys, and Wein [HSW96] showed very recently that an optimal fractional solution of the linear relaxation of a time-indexed integer programming formulation of the single-machine total weighted completion time problem with precedence constraints can be rounded to a schedule of objective function value at most 4 times as large. Unfortunately, because of the time indices this formulation is only pseudo-polynomial in the input size. Since their algorithm relies on solving the LP relaxation this algorithm is only pseudopolynomial as well. However, Hall, Shmoys, and Wein present a new linear programming formulation of polynomial size, by relying on time intervals instead of single points in time, which leads to an algorithm guaranteed to produce a schedule with total weighted completion time at most (4 + e) of the optimal value, for any fixed e > 0. When incorporating release dates, their algorithm produces a solution at most a factor of (3 + 2x/2 + e) worse than the optimal one. If there are release dates but no precedence constraints, they are able to turn it into a 4-approximation algorithm. Up to now, these have been the best bounds obtained; see [HSW96] for the short history. Our work is strongly motivated by the work of Hall, Shmoys, and Wein (but is not based on their results), and the observations Maurice Queyranne and the author made comparing different polyhedral formulations of scheduling problems, see [QS94]. We also refer the reader to [QS94] for an overview on polyhedral approaches to scheduling. We present much simpler algorithms using the optimal solution to a linear programming relaxation in completion time (or natural date) variables that achieve a performance ratio of (2 - ~-~) 2 for the single-machine problem subject to precedence constraints, and of 3 for the single-machine problem subject to precedence constraints and release dates (Sect. 2). This not only significantly improves the best performance guarantees known but provides new insight into the strength of several related LP relaxations of these problems. In particular, it implies that both the linear programming relaxation in time-indexed variables as well as in linear ordering variables can be used to obtain an approximation algorithm with the same performance ratio for the first problem. Moreover, by proposing new linear programming relaxations for identical par-
303
allel machine problems (Sect. 3) as well as uniform parallel machine problems (Sect. 4) we extend our method to these models as well. This leads, for instance, to a (4 - -~)-approximation algorithm for identical machines and release dates. This algorithm is slightly better than the (4 + e)-approximation algorithm that is the best previously known [HSW96] and is much simpler. We also extend these techniques to the problem of minimizing the total weighted completion time subject to both release dates and precedence constraints in a flow shop and develop a simple (2 m + 1)-approximation algorithm in this case. Finally, we present a 3 m-approximation algorithm for multi-stage parallel processor flow shop scheduling to minimize the total weighted completion time (Sect. 5). Tables 1 - 5 summarize the results of this paper, and compare the bounds with the best previously known, if any. Single Machine
Known
Precedence Constraints
New
4 + e
Release Dates Precedence Constraints and Release Dates
2
2 a-l-1
4
3
3 + 2 v~ + e
3
T a b l e 1: S u m m a r y of r e s u l t s for t h e m i n i m i z a t i o n of t h e t o t a l w e i g h t e d c o m p l e t i o n t i m e on a single m a c h i n e . T h e " K n o w n " c o l u m n lists t h e b e s t k n o w n p r e v i o u s b o u n d s , w h i l e t h e " N e w " c o l u m n lists n e w r e s u l t s f r o m t h i s p a p e r . All t h e b e s t k n o w n b o u n d s a r e d u e t o H a l l , S h m o y s , a n d W e i n [ H S W 9 6 ] . N o t e t h a t for t h e m o d e l s w i t h p r e c e d e n c e c o n s t r a i n t s t h i s w e r e t h e b e s t k n o w n b o u n d s for t h e m i n i m i z a t i o n of t h e t o t a l ( u n w e i g h t e d ) c o m p l e t i o n t i m e as well.
Identical Parallel Machines
Known v~+l
Release Dates
4 4- e
New
(3 -
~)(1
-
4
! r~
~-~-r)
T a b l e 2; S u m m a r y of r e s u l t s for t h e m i n i m i z a t i o n of t h e t o t a l w e i g h t e d c o m p l e t i o n t i m e on i d e n t i c a l p a r a l l e l m a c h i n e s . T h e first k n o w n b o u n d is d u e t o K a w a g u c h i a n d K y a n [KK86]; t h e s e c o n d is d u e t o H a l l , S h m o y s , a n d W e i n [ H S W 9 6 ] . A l t h o u g h t h e f o r m e r is b e t t e r t h a n o u r s , we h a v e i n c l u d e d it s i n c e it s h o w s w h a t c a n b e a c h i e v e d w i t h o u r m e t h o d . I n d e e d , o u r a p p r o a c h h a s t h e a d v a n t a g e of b e i n g a b l e to h a n d l e q u i t e n a t u r a l l y a d d i t i o n a l r e s t r i c t i o n s like r e l e a s e d a t e s a n d p r e c e d e n c e c o n s t r a i n t s . In a d d i t i o n , a t t h e s a m e t i m e it gives t h e s a m e g u a r a n t e e for t h e L P l o w e r b o u n d . Here, m d e n o t e s t h e n u m b e r of machines.
Uniform Parallel Machines
Known 4 + ~
Release Dates
4 + e
New m i n { 2 -k ( m - - 1) ~ , m i n { 3 + ( m - 1) . . . . . ~ '
2
}
1 q- 2 ~ I max s~
T a b l e 3: S u m m a r y of r e s u l t s for t h e m i n i m i z a t i o n of t h e t o t a l w e i g h t e d c o m p l e t i o n t i m e o n u n i f o r m p a r a l l e l m a c h i n e s . H e r e , sl d e n o t e s t h e s p e e d of m a c h i n e i. B o t h b e s t k n o w n b o u n d s a r e d u e t o H a l l , S h m o y s , a n d W e i n [ H S W 9 5 ] . In t h e first c a s e o u r b o u n d is n o t w o r s e t h a n 1 4- ~ - 1. H e n c e it is a l w a y s b e t t e r t h a n t h e k n o w n o n e for m < 5. In case of r e l e a s e d a t e s o u r b o u n d is n o t w o r s e t h a n 2 4- 2 x / ~ -- 1. T h e r e f o r e it is a l w a y s b e t t e r t h a n t h e k n o w n o n e for m 0. Moreover, our algorithm seems to be more natural and simpler as they need to sort the
308
jobs based on the time by which half of their processing has been completed in the fractional interval-indexed solution. Finally, note that we do not rely on the ellipsoid method for solving the linear programming relaxation in the linear ordering variables. We now turn to the situation where jobs arrive over time, which is another characteristic encountered frequently in real-world problems. In addition to the setting described above, the jobs are now assumed to be released at different times, where job j is released at time rj >_ O. We show that algorithm 1NATURAL when given the optimal solution to a slightly extended linear program produces a solution within a factor less than 3 of the optimal. We consider the following formulation: minimize C wjCj j----1 j~A
for all A C_ N,
+ jeA ZpJ
jEA
G >Cj+pk
for j -+ k,
Cj >_ rj + pj
for all j E N.
(3)
L e m m a 4 . Let C be any feasible solution to the linear relaxation (3). Assume, w. I. o. g., that C1 _Chj+Phk
j--+k,
h=l,...,m,
Vii >_rj + Plj 9 We call the flow shop heuristic F-NATURAL. Find an optimal solution C LP to (8). Assume that C LP _ CLmP < . " < CLmP, and schedule the jobs in this order, without unforced idle time. Thus, CHj = max{C~h_l) j +Phj, cH(j_I)+Phi} where we assume, to simplify notation, that C0~ = rj and CHo = 0. In particular, the solution constructed this way is a permutation schedule. Since m
j
h=2 k=l
the following theorems can be proved by multiple applications of Lemma 1.
313
T h e o r e m 10. For F]prec[ ~ wjCj, let w(F-•ATURAL) be the value of the schedule produced by algorithm F-NATURAL, and let w(OPT) be the value of an optimal schedule. Then w(F-NATURAL)
~
(2m
n-2~1) w(OPT) .
In the presence of release dates we have to apply Lemma 4 for the first machine. T h e o r e m 11. For F[rj, prec I ~ wjCj, let w(F-NATURAL) be the value of the schedule produced by algorithm F-NATURAL, and let w(OPT) be the value of an optimal schedule. Then w(F-NATURAL)
~
(2m + 1) w(OPT) .
Again, we also derive the respective guarantees for the lower bounds obtained from solving the linear programming problem (8). In addition, our analysis carries over to no-wait flow shop problems (see [Sch95a, Sch95b] for details). We close this section by giving a sketch of the results for multiprocessor flow shop scheduling. This is a generalization of the usual flow shop in which each machine is replaced by a set of identical parallel machines (cf., e. g., [RC92, HHLV]). If we combine the parallel machine inequalities (4) in a similar manner as we combined the single machine inequalities in (8) we can obtain an approximation algorithm with performance guarantee 3m (or 3 m + 1, in case of job release dates). Here, m is the number of stages. 6
Concluding
Remarks
With this work we continue a series of recent papers on provably good approximation algorithms based on "rounding" optimal solutions to LP relaxations of scheduling problems to feasible solutions; see, in particular, [LST90, ST93, PSW95, HSW96]. Thereby, the results obtained prove that linear relaxations are a quite powerful tool for effectively producing not only lower bounds but also good feasible solutions. They also prove the strength of polyhedral formulations that are based on completion time variables. This fits nicely with the observations that, in the absence of precedence constraints, Smith's rule (cf., [Smi56]) can be obtained from this formulation, and that complete linear descriptions of the convex hull of feasible completion time vectors are also known when the additional restrictions have a special structure that admits polynomial optimization, see [QS94, QS95, Sch95a] for details. Moreover, most of the approximation algorithms presented here are not only better than the comparable ones of Hall, Shmoys, and Wein, they seem also to be simpler, both in terms of explanation and analysis. Indeed, the techniques we use are not too sophisticated. The results simply document the strength of the LP relaxations. It is obvious that better knowledge of the underlying P01yhedra, i. e., more efficiently solvable classes of valid inequalities, may not only lead to stronger lower bounds but also to better approximation algorithms.
314
Acknowledgements. The author is grateful to Maurice Queyranne and David Shmoys for helpful discussions and comments on the text, to Prancois Margot for helpful comments, and to Leslie Hall and Joel Wein for helpful discussions. The author has been supported by the graduate school "Algorithmische Diskrete M a t h e m a t i k " . The graduate school "Algorithmische Diskrete Mathematik" is supported by the Deutsche Forschungsgemeinschaft (DFG), grant We 1265/2-1.
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