J. Lenstra, A. Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343-362, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Gross [21] 1jp i = 1; prec(l ij )jCmax Palem and Simons [34] 1jp i = 1; chains(l ij 2 f0; k1 ; k2g)jCmax P jp i = 1; precjCmax P jjCmax Ullman [42] P2jp i 2 f1; 2g; precjCmax Brucker and Knust [7] 1jp i = 1; intree(l ij = l) r i jCmax 1jp i = 1; outtree(l ij = l)jLmax Lenstra et al. [29] 1jr i jLmax Fig. 6. NP hardness reductions. All problems listed are known to be NP hard. runs in O(ne e log n) where e is number of edges after transitive closure. The latter algorithm runs in O(n ) excluding the time to perform transitive reduction. 3) Moving to the third group, ....

....to 1 and 2, the problem is NP hard on two processors in the presence of precedence constraints. However, further restricting the problem to unit processing time makes it trivially polynomially decidable. Moving onto the case when all precedence constraints are eliminated. Lenstra et al. [29] show that the one processor scheduling problem with individual processing times and in the presence of release times and deadlines, is NP hard. However, when the release times are also omitted, the problem becomes polynomially solvable with Lawler s algorithm [28] Similarly, when all processing ....

Lenstra, J. K., Kan, A. H. G. R., and Brucker, P. Complexity of machine scheduling problems. Annals of Discrete Mathematics 1 (1977), 343-362.

....it uses. Mok [8] has shown that the problem of deciding whether it is possible to schedule a set of periodic processes, which use semaphores to enforce mutual exclusion, is NP hard. Indeed, most scheduling problems for processes that have time requirements and mutual exclusive resources am NP hard [29, 30]. Similarly, the analysis of a concurrent program, which uses arbitrary interprocess message passing (synchronous or asynchronous) appears to be computationally intractable. This does not imply that it is impossible to construct polynomial time feasibility tests but that necessary and suffi cient ....

....aperiodic processes, copies of the code of the process are held at nodes that are likdy to be asked to guarantee the process. 4. 3 Remote blocking If we now move to consider process interaction in a multiprocessor system, then the complexity of the scheduling is further increased (i.e. NP hard) [29, 30, 54, 55]. As with the uniprocessor discussion, we restrict ourselves to the interactions that take the form of mutual exclusive synchronisation for controlling resource usage. In the dynamic (flexible) system described above, in which aperiodic processes are moved in order to find a node that will ....

LENSTRA, J.K., RINNOOY, A.H.G., and BRUCKER, P.: 'Complexity of machine scheduling problems', Ann. Discrete Math., 1977, 1

....integration makes the problem of scheduling for an FMS more difficult. Therefore, effective scheduling schemes are required to maximize system throughput. The problem of finding an optimal schedule for the manufacture of a set of products with multiple possible routings is known to be NP hard [LRB77]. Mathematical techniques like branch and bound and dynamic programming techniques may require exponential time in the worst case. Polynomial time approximation schemes for solving scheduling problems which produce sub optimal schedules within polynomial time have been examined. These techniques ....

J. Lenstra, A. Rinnooy and P. Bruckner, "Complexity of Machine Scheduling Problems", Annals of Discrete Mathematics , Vol. 7, 1977.

....denoted by C j . We aim to minimize the total weighted completion time j2J w j C j where w j denotes a given nonnegative integral weight of job j which is a measure for its importance. In the standard notation of Graham et al. 16] this problem reads R j j w j C j . It is shown to be NP hard in [3, 26], even for a fixed number m 2 of identical parallel machines. The special case of identical parallel machines, i.e. for each job j and all machines i we have p i j = p j , is denoted by P j j w j C j . The corresponding single machine scheduling problem can be solved in polynomial time using ....

J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343 -- 362, 1977.

....of maximum and minimum job processing times, and n and m denote the number of jobs and machines respectively. This is also the best offline approximation algorithm for the problem. In contrast to the unweighted problem, the problem with weights is known to be strongly NP hard on a single machine [9]. Until recently no non trivial approximation algorithms or online algorithms were known for this problem. Chekuri, Khanna, and Zhu [4] very recently obtained several results. In particular they give a semionline algorithm for a single machine that is O(log 2 P ) competitive. The algorithm is ....

....and excluded time intervals. We can now invoke Theorem 3 to compute (1 ffl) approximate schedules in polynomial time. 4 A PTAS for Bounded P In this section we give a PTAS for the case when W is unrestricted but P is bounded. A slight modification to the NP hardness for arbitrary P and W [9] shows that the problem remains strongly NP hard for any fixed P 1 and bounded W . In fact it is not known whether the case of P = 1 (that is all processing times are the same) is polynomial time solvable or not Our PTAS for this case is based on G r a prioritized schedules where the ....

J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343--362, 1977.

.... also been derived by Jay Sethuraman and Mark Squillante [48] Unfortunately, for the general network scheduling problem including release dates the situation is more complicated; for a given assignment of jobs to machines, the sequencing problem on each machine is still strongly NP hard, see [29]. However, we know that in an optimal schedule a violation of Smith s ratio rule can only occur after a new job has been released; in other words, whenever two successive jobs on machine i can be exchanged without violating release dates, the job with the higher ratio w j #p ij will be processed ....

....finding an optimal sequence for the jobs on the machines is easy. Unfortunately, for the general network scheduling problem including release dates the situation is more complicated; for a given assignment of jobs to machines, the sequencing problem on each machine is still strongly NP hard, see [29]. 3.1 Scheduling in time slots Notice, however, that in an optimal schedule a violation of Smith s ratio rule can only occur after a new job has been released; in other words, whenever two successive jobs on machine i can be exchanged without violating release dates, the job with higher ratio ....

J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343 -- 362, 1977.

....(where C i denotes the completion time of task i) Following the classical scheduling notation (see for instance [2, 3] the corresponding scheduling problems are referred to as P jsize i jC max and P jsize i j P C i . Scheduling jobs to minimize makespan is NP Hard even on 2 parallel machines [9] so, the more complex problem P2jsize i jC max is also NP Hard. Du and Leung [6] have shown that P5jsize i jC max is binary NP Hard. The same authors have also proposed a pseudopolynomial time algorithm for the 2 and 3 processors case. When the number of processors is not xed (i.e. m is a data) ....

J. K. Lenstra, A. H. G. Rinnooy Kan and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics 1:343{ 362, 1977.

....colony of ants and is highly effective in finding the best known solutions on all instances of a widely used set of benchmark problems. 1 Introduction In this paper we study the single machine total weighted tardiness problem (SMTWTP) 21] a scheduling problem that is known to be NP hard [16] and for which instances with more than 50 jobs can often not be solved to optimality with state of the art branch bound algorithms [1, 5] In the SMTWTP n jobs have to be sequentially processed on a single machine. Each job j has a processing time p j , a weight w j , and a due date d j ....

J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problem. In P. L. Hammer, E. L. Johnson, B. H. Korte, and Nemhauser G. L., editors, Studies in Integer Programming, volume 1 of Annals of Discrete Mathematics, pages 343--362. North-Holland, Amsterdam, NL, 1977.

....search algorithm, the use of candidate lists that guide the ants solution construction, and a heterogeneous ant colony where ants apply various local search variants concurrently. 1 Introduction The single machine total weighted tardiness problem (SMTWTP) is an NP hard scheduling problem [9] for which instances with more than 50 jobs often can not be solved to optimality with state of the art branch and bound algorithms [1] Hence, to solve large problem instances one needs approximation algorithms which obtain near optimal solutions in short time. Recently, algorithms falling ....

J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Bruckner. Complexity of machine scheduling problems. In P. L. Hammer, E. L. Johnson, B. H. Korte, and G. L. Nemhauser, editors, Studies in Integer Programming, volume 1 of Annals of Discrete Mathematics, pages 343--362. North-Holland, Amsterdam, NL, 1977.

....p i = p; r i j P w i U i ) Both problems being NP Hard in the general case. Simons ( 11] provides a polynomial algorithm to minimize the sum of the completion times when processing times are equal (P jp i = p; r i j P C i ) while 2 the simple problem (1jr i j P C i ) is NP Hard [10]. We show that for each xed value of m, the problem with equal processing times is solvable in polynomial time provided that Hypothesis 1 holds: Hypothesis 1 The functions f i are non decreasing, i.e. 8t 1 ; 8t 2 t 1 ; f i (t 1 ) f i (t 2 ) and the functions f i f j are monotonous, i.e. ....

Jan Karel Lenstra, Alexander H.G. Rinnooy Kan and Peter Brucker, Complexity of machine scheduling problems, Annals of Discrete Mathematics, 1 (1977) 343-362.

....pseudopolynomially solvable special case with different release dates. L2kC max remains open for the ordinary or strong NP hardness. The NP hardness of L2kC max , C2jp ij 2 f1; 2gjC max , 1Cjn = 1; p i = 1jC max and R1F jn = 1; p o ]jP strengthens earlier NP hardness results of Lenstra et al. [LRKB77] for J2jm j 3jC max , Lenstra and Rinnooy Kan [LRK79] for J2jp ij 2 f1; 2gjC max , Roundy [R92] for Cjn = 1jC max , and Livshits et al. LMC74] for R1F jn = 1; p o ]jP , respectively. Note that the NP hardness proof of Hall et al. HLP97] for J2jm j = 3jB max fits for L2kB max in fact. All the ....

J. K. Lenstra, A. H. G. Rinnooy Kan and P. Brucker, Complexity of machine scheduling problems, Annals of Discrete Math. 1 (1977), 363--362.

....that schedules the job with the largest weight among the alive jobs has a competitive ratio of Theta(n) Results: In this paper we address the weighted flow time problem. In contrast to the unweighted case, the problem with weights is known to be strongly NP hard even on a single machine [9]. We give a semi online algorithm for a single machine that is O(log 2 P) competitive. The algorithm is semi online in that the parameter P , the ratio of the maximum job size to the minimum job size, is known in advance. It uses this information only to round the weights of the jobs in a ....

J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343--362, 1977.

....136, 10623 Berlin, Germany. Email: skutella math.tu berlin.de PeopleSoft Inc. San Mateo, CA, USA 94404. Email: Yaoguang Wang peoplesoft.com 1 In the classical scheduling notation [12] this problem is denoted by 1j r j j P w j C j . It is strongly NP hard, even if w j = 1 for all jobs j [17]. One of the key ingredients in the design and analysis of approximation algorithms as well as in the design of implicit enumeration methods is the choice of a bound on the optimal value. Several linear programming based as well as combinatorial lower bounds have been proposed for this well ....

J.K. Lenstra, A.H.G. Rinnooy Kan and P. Brucker, \Complexity of machine scheduling problems", Annals of Discrete Mathematics, 1, 343-362 (1977).

....scheme (PTAS) for this problem. Our ideas also extend to the preemptive case Qjr j ; pmtnj P j w j C j but we omit the details of that result in this extended abstract. Most variants of scheduling to minimize average completion time are strongly NP hard including preemptive problems [12]. Polynomial time solvable cases include P jj P j C j , 1jj P j w j C j , and Rjj P j C j . In the last few years considerable progress has been made in understanding the approximability of many of these NP hard problems. Constant and logarithmic ratio approximations were found for several ....

J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343--362, 1977.

....pseudopolynomially solvable special case with di erent release dates. L2kC max remains open for the ordinary or strong NP hardness. The NP hardness of L2kC max , C2jp ij 2 f1; 2gjC max , 1Cjn = 1; p i = 1jC max and R1F jn = 1; p o ]jP strengthens earlier NP hardness results of Lenstra et al. [LRKB77] for J2jm j 3jC max , Lenstra and Rinnooy Kan [LRK79] for J2jp ij 2 f1; 2gjC max , Roundy [R92] for Cjn = 1jC max , and Livshits et al. LMC74] for R1F jn = 1; p o ]jP , respectively. Note that the NP hardness proof of Hall et al. HLP97] for J2jm j = 3jB max ts for L2kB max in fact. All the ....

J. K. Lenstra, A. H. G. Rinnooy Kan and P. Brucker, Complexity of machine scheduling problems, Annals of Discrete Math. 1 (1977), 363-362. ON SCHEDULING CYCLE SHOPS 33

....to an instance of 1jr j j L max . Consequently, in the remainder of the lecture we will focus on 1jr j j L max . The reduction we have given from 1jr j jL max to 1jr j j L max establishes that 1jr j j L max is weakly NP hard. In fact, 1jr j j L max is strongly NP hard ([2]) A survey of results on this problem appears in [3, section 1.2] 2.3 Jackson s rule for 1jr j j L max Let us now modify the EDD rule to the setting of 1jr j j L max . We do so by considering the reduction from 1jr j jL max to 1jr j j L max . An early due date 2 d k corresponds ....

J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics 1 (1977), 343--362.

....second column are NP hard or strongly NP hard as it has been established in the papers, references to which are listed in the rst column. Table 4. Reductions proving the NP hardness [DLY90] P2jpmtn; r j j C j J2jr j ; p ij = 1j C j [DLW92] P2jpmtn; r j j U j J2jr j ; p ij = 1j U j [BCS74] [LRKB77] P2jpmtnj w j C j J2jp ij = 1j w j C j [K72] 1jpmtnj w j U j J2jp ij = 1j w j U j [DL90] 1jpmtnj T j J2jp ij = 1j T j [LRKB77] P2kC max J2jno wait; p ij = 1jC max [BCS74] LRKB77] P2k w j C j J2jno wait; p ij = 1j w j C j Table 5. Reductions proving the strong NP hardness [DLY91] ....

....Table 4. Reductions proving the NP hardness [DLY90] P2jpmtn; r j j C j J2jr j ; p ij = 1j C j [DLW92] P2jpmtn; r j j U j J2jr j ; p ij = 1j U j [BCS74] LRKB77] P2jpmtnj w j C j J2jp ij = 1j w j C j [K72] 1jpmtnj w j U j J2jp ij = 1j w j U j [DL90] 1jpmtnj T j J2jp ij = 1j T j [LRKB77] P2kC max J2jno wait; p ij = 1jC max [BCS74] LRKB77] P2k w j C j J2jno wait; p ij = 1j w j C j Table 5. Reductions proving the strong NP hardness [DLY91] P2jpmtn; chainsj C j J2jchains; p ij = 1j C j [LLLRK84] 1jpmtn; r j j w j C j J2jr j ; p ij = 1j w j C j [L77] LRKB77] 1jpmtnj w ....

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J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker, Complexity of machine scheduling problems, Ann. Discrete Math. 1 (1977), 343-362.

....at time zero. The tardiness of a job j is defined as T j = maxf0; C j d j g, where C j is the completion time of job j in the current job sequence. The goal is to find a job sequence which minimizes the sum of the weighted tardiness given by P n i=1 w i T i . The SMTWTP is an NP hard [9] scheduling problem and instances with more than 50 jobs can often not be solved to optimality with state of the art branch bound algorithms [1, 4] Therefore, several heuristic methods have been proposed for its solution. These include simple construction heuristics like the Earliest Due Date ....

J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problem. In P. L. Hammer et al., editor, Studies in Integer Programming, volume 1 of Annals of Discrete Mathematics, pages 343--362. North-Holland, Amsterdam, NL, 1977.

....this approach might well prove useful in a more general context. Unfortunately, for the general network scheduling problem including release dates the situation is more complicated; for a given assignment of jobs to machines, the sequencing problem on each machine is still strongly NP hard, see [Lenstra et al. 1977]. However, we know that in an optimal schedule a violation of Smith s ratio rule can only occur after a new job has been released; in other words, whenever two successive jobs on machine i can be exchanged without violating release dates, the job with the higher ratio w j =p ij will be ....

....of nding an optimal sequence for the jobs on the machines is easy. Unfortunately, for the general network scheduling problem including release dates the situation is more complicated; for a given assignment of jobs to machines, the sequencing problem on each machine is still strongly NP hard, see [Lenstra et al. 1977]. Convex Quadratic and Semide nite Programming Relaxations in Scheduling 15 1 performance ratio c T a Z CQP 0 ( a) 1 0 3 2 Fig. 1. The performance of Randomized Rounding depends on c T a Z CQP 0 ( a) 3.1 Scheduling in time slots Notice, however, that in an optimal schedule a ....

Lenstra, J. K., Rinnooy Kan, A. H. G., and Brucker, P. 1977. Complexity of machine scheduling problems. Annals of Discrete Mathematics 1, 343 - 362.

....a specified release date r j before which it may not begin processing, j = 1, n; furthermore, they consider the unit weight case, or in other words, w j = 1, for each j = 1, n. This problem is denoted 1 r j # C j and was shown to be NP hard by Lenstra, Rinnooy Kan, Brucker [22]. The algorithm of Phillips, Stein, Wein [29] is based on a relaxation of the problem that can be solved in polynomial time. In this case, however, the relaxation is not a linear program, but instead one motivated in purely scheduling terms: rather than requiring that each job be processed ....

J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Ann. Discrete Math., 1:343--362, 1977.

....only takes period into account. Third, the schedulability analysis of DM is well known [3] although not optimal in a distributed system [9] In fact, it has been shown that the problem of scheduling any nontrivial system of tasks requiring ordered execution on more than two processors is NP hard [10]. Distributed Priority Ceiling. In our scheduling approach, we use the distributed priority ceiling protocol (DPCP) for resource access, such as the access of servers by clients. In a single node system, schedulability of hard real time tasks that require resources can be computed using ....

J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of Machine Scheduling Problem. Annals of Discrete Mathematics, 1:343-362, 1977.

....objective is to minimize the number of tardy jobs, NTJ(I) P i2I U i . In the sequels, D i = r i ; d i ] denotes the time window of job J i , that is to say the time interval in which J i must execute if it is on time. With arbitrary release dates, the problem is NP Hard in the strong sense [LRKB77]. However, many special cases and or relaxations of the problem are polynomially solvable. With equal release dates, the problem 1jj P U i is polynomially solvable [MOO68] Lawler [LAW90] has proposed an O(n 5 ) dynamic programming algorithm for the preemptive case 1jr i ; pmtnj P U i ....

J.K. LENSTRA, A.H.G. RINNOOY KAN, and P. BRUCKER. Complexity of machine scheduling problems. Annals of Discrete Mathematics, (1):343--362, 1977.

....are compatible with the machine speeds. However, if we allow for arbitrary processing times (in case m 2) or arbitrary release dates, a complete linear description may be much more difficult to obtain since the problems P2j j P w j C j and 1jr j j P w j C j are already NP hard (cf. 5] and [12]) However, the inequalities presented above can be extended to more complicated problems. For example, consider the single machine problem with arbitrary processing times and release dates. The following inequality [17] is a natural combination of the ideas in [18] and in Sect. 4 above: X j2A ....

J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker, Complexity of machine scheduling problems, Annals of Discrete Mathematics 1 (1977), 343--362.

.... ij ) Cmax Palem and Simons [29] 1 p i = 1; chains(l ij # 0, k1 , k2 ) Cmax Garey and Johnson [11] P p i = 1; prec Cmax P Cmax Ullman [36] P2 p i # 1, 2 ; prec Cmax Brucker and Knust [3] 1 p i = 1; intree(l ij = l) r i Cmax 1 p i = 1; outtree(l ij = l) Lmax Lenstra et al. [24] 1 r i Lmax Fig. 6. NP hardness reductions. All problems listed are known to be NP hard. scheduling to minimize the makespan on one processor with arbitrary precedence constraints, individual processing times p i # 1 and zero or unit latencies; the work in [6] proves this fact for unit ....

....on two processors in the presence of precedence constraints. However, further restricting the problem to unit processing time makes it trivially polynomially decidable. Moving onto the case when all precedence constraints are eliminated. Lenstra et Scheduling Time Constrained Instructions 9 al. [24] show that the one processor scheduling problem with individual processing times and in the presence of release times and deadlines, is NP hard. However, when the release times are also omitted, the problem becomes polynomially solvable with Lawler s algorithm [23] Similarly, when all processing ....

Lenstra, J. K., Kan, A. H. G. R., and Brucker, P. Complexity of machine scheduling problems. Annals of Discrete Mathematics 1 (1977), 343--362.

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J. Lenstra, A. Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343-362, 1977.

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Lenstra, J.K., Kan, A.H.G.R., Brucker, P.: Complexity of machine scheduling problems. Annals of Discrete Mathematics 1 (1977) 343-362

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J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343-362, 1977.

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J.K.Lenstra, A. Kan, and P.Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343--362, 1977.

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J.K. Lenstra, A.H.G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343-362, 1977.

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J.K. Lenstra, A.H.G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343--362, 1977.

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J.K. Lenstra, A.H.G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343-362, 1977.

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J.K. Lenstra, A.H.G. Rinnooy Kan and P. Brucker, Complexity of machine scheduling problems, Annals of Discrete Mathematics 1, (1977), 343-362.

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J. K. Lenstra, A. H. G. Rinnoy Kan and P. Brucker. Complexity of machine scheduling problems. Ann. Discrete Math. 1, 1977.

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J. Lenstra, A. Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343--362, 1977.

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J. Lenstra, A. Rinnooy, and P. Brucker, "Complexity of machine scheduling problems," Annals of Discrete Mathematics, Vol. 1, 1977.

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J.K.Lenstra, A. Kan, and P.Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343--362, 1977.

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J. K. Lenstra, R. Kan, A. H. G., and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Machines, 1:343--362, 1977.

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J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker, Complexity of machine scheduling problems, Annals of Discrete Mathematics, 343-351 (1977).

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Lenstra, J.K., Rinnooy Kan, A.H.G. and Brucker, P. (1977). Complexity of Machine Scheduling Problems. Annals of Discrete Mathematics, 1 pp 343-362.

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J. Lenstra, A. Rinnooy, and P. Brucker. Complexity of Machine Scheduling Problems. Annals of Discrete Mathematics, 1, 1977.

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Jan Karel Lenstra, Alexander H.G. Rinnooy Kan, and Peter Brucker. Complexity of machine scheduling problems. Ann. Discrete Math., 1:343--362, 1977.

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J.K. Lenstra, A.H.G. Rinnooy Kan, P. Brucker (1977) Complexity of machine scheduling problems, Annals of Discrete Mathematics 1, 343-362.

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LENSTRA, J. K., KAN, A. H. G. R., AND BRUCKER, P. Complexity of machine scheduling problems. Annals of Operation Research 4 (1977), 343--362.

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J. Lenstra, A. Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343-362, 1977.

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J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343-362, 1977.

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J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problem. In P. L. Hammer et al., editor, Studies in Integer Programming, volume 1 of Annals of Discrete Mathematics, pages 343--362. North-Holland, Amsterdam, NL, 1977.

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J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343362, 1977.

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J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Bruchker, "Complexity of machine scheduling problems," Annals of Discrete Mathematics, vol. 1, pp. 343--362, 1977.

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Lenstra, J K, Rinnooy Kan, A H G and Brucker, P (1977). Complexity of Machine Scheduling Problems, Annals of Discrete Mathematics, Vol.1, 343-62.

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J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1:343--362, 1977.

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