Abstract
In flow shop scheduling there are m machines and n jobs, such that every job has to be processed on the machines in the fixed order 1,...,m. In the permutation flow shop problem, it is also required that each machine processes the set of all jobs in the same order. Formally, given n jobs along with their processing times on each machine, the goal is to compute a single permutation of the jobs σ:[n]→[n], that minimizes the maximum job completion time (makespan) of the schedule resulting from σ. The previously best known approximation guarantee for this problem was \(O(\sqrt{m\log m})\) [29]. In this paper, we obtain an improved \(O(\min\{\sqrt{m},\sqrt{n}\})\) approximation algorithm for the permutation flow shop scheduling problem, by finding a connection between the scheduling problem and the longest increasing subsequence problem. Our approximation ratio is relative to the lower bounds of maximum job length and maximum machine load, and is the best possible such result. This also resolves an open question from [21], by algorithmically matching the gap between permutation and non-permutation schedules. We also consider the weighted completion time objective for the permutation flow shop scheduling problem. Using a natural linear programming relaxation, and our algorithm for the makespan objective, we obtain an \(O(\min\{\sqrt{m},\sqrt{n}\})\) approximation algorithm for minimizing the total weighted completion time, improving upon the previously best known guarantee of εm for any constant ε> 0 [30]. We give a matching lower bound on the integrality gap of our linear programming relaxation.
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References
Chakrabarti, S., Phillips, C., Schulz, A., Shmoys, D., Stein, C., Wein, J.: Improved Scheduling Algorithms for Minsum Criteria. In: Meyer auf der Heide, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 646–657. Springer, Heidelberg (1996)
Chazelle, B.: The discrepancy method. Randomness and complexity. Cambridge University Press, Cambridge (2000)
Chen, B., Potts, C., Woeginger, G.: A review of machine scheduling: complexity, algorithms and approximability. In: Du, D.-Z., Pardalos, P.M. (eds.) Handbook of combinatorial optimization, vol. 3, pp. 21–169. Kluwer Academic Publishers, Boston (1998)
Conway, R., Maxwell, W., Miller, L.: Theory of scheduling. Addison-Wesley Publishing Co., Reading, Mass, London, Don Mills, Ont. (1967)
Czumaj, A., Scheideler, C.: A New Algorithmic Approach to the General Lovasz Local Lemma with Applications to Schedulung and Satisfiability Problems. In: Proc. 32 ACM Symposium on Theory of Computing (STOC) (2000)
Feige, U., Scheideler, C.: Improved bounds for acyclic job shop scheduling. In: STOC 1998. Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pp. 624–633. ACM Press, New York (1998)
Fishkin, A., Jansen, K., Mastrolilli, M.: On minimizing average weighted completion time: a PTAS for the job shop problem with release dates. In: Ibaraki, T., Katoh, N., Ono, H. (eds.) ISAAC 2003. LNCS, vol. 2906, pp. 319–328. Springer, Heidelberg (2003)
Framinan, J., Gupta, J., Leisten, R.: A review and classification of heuristics for permutation flow-shop scheduling with makespan objective. Journal of the Operational Research Society 55, 1243–1255 (2004)
Frieze, A.: On the length of the longest monotone subsequence of a random permutation. The Annals of Applied Probability 1(2), 301–305 (1991)
Hall, L.A., Shmoys, D.B., Wein, J.: Scheduling to Minimize Average Completion Time: Off–line and On–line Algorithms. In: Proceedings of the 7th Symposium on Discrete Algorithms, pp. 142–151 (1996)
Hall, L.A., Schulz, A.S., Shmoys, D.B., Wein, J.: Scheduling to Minimize Average Completion Time: Off–Line and On–Line Approximation Algorithms. Mathematics of Operations Research 22, 513–544 (1997)
Jansen, K., Solis-Oba, R., Sviridenko, M.: Makespan Minimization in Job Shops: a Linear Time Approximation Scheme. SIAM Journal of Discrete Mathematics 16, 288–300 (2003)
Johnson, S.: Optimal two- and three-stage production schedules with setup times included. Naval Research Logistics Quartely 1, 61–68 (1954)
Hofri, M.: Probabilistic Analisys of Algorithms: On Computing Metodologies for Computing Algorithms Performance Evaluation. Springer, Heidelberg (1987)
Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B.: Sequencing and scheduling: Algorithms and complexity. In: Handbook in Operations Research and Management Science, vol. 4, pp. 445–522. North-Holland, Amsterdam (1993)
Logan, B.F., Shepp, L.A.: A Variational Problem for Random Young Tableaux. Advances in Mathematics 26, 206–222 (1977)
Nawaz, M., Enscore Jr., E., Ham, I.: A heuristic algorithm for the m-machine n-job flow-shop sequencing problem. OMEGA International J. Management Sci. 11, 91–95 (1983)
Nowicki, E., Smutnicki, C.: New results in the worst-case analysis for flow-shop scheduling. Discrete Appl. Math. 46, 21–41 (1993)
Nowicki, E., Smutnicki, C.: Worst-case analysis of an approximation algorithm for flow-shop scheduling. Oper. Res. Lett. 8, 171–177 (1989)
Nowicki, E., Smutnicki, C.: Worst-case analysis of Dannenbring’s algorithm for flow-shop scheduling. Oper. Res. Lett. 10, 473–480 (1991)
Potts, C., Shmoys, D., Williamson, D.: Permutation vs. nonpermutation flow shop schedules. Operations Research Letters 10, 281–284 (1991)
Queyranne, M.: Structure of a simple scheduling polyhedron. Math. Programming Ser. A 58(2), 263–285 (1993)
Queyranne, M., Sviridenko, M.: Approximation Algorithms for Shop Scheduling Problems with Minsum Objective. Journal of Scheduling 5, 287–305 (2002)
Raghavan, P.: Probabilistic construction of deterministic algorithms: approximating packing integer programs. J. Comput. System Sci. 37, 130–143 (1988)
Röck, H., Schmidt, G.: Machine aggregation heuristics in shop-scheduling. Methods of Operations Research 45, 303–314 (1983)
Sevast’janov, S.: On some geometric methods in scheduling theory: a survey. Discrete Applied Mathematics 55, 59–82 (1994)
Schrijver, A.: Combinatorial optimization. Polyhedra and efficiency. In: Algorithms and Combinatorics, vol. B 24. Springer, Berlin (2003)
Shmoys, D., Stein, C., Wein, J.: Improved Approximation Algorithms for Shop Scheduling Problems. SIAM Journal on Computing 23(3), 617–632 (1994)
Sviridenko, M.: A Note on Permutation Flow Shop Problem. Annals of Operations Research 129, 247–252 (2004)
Smutnicki, C.: Some results of the worst-case analysis for flow shop scheduling. European Journal of Operational Research 109, 66–87 (1998)
Vershik, A.M., Kerov, S.V.: Asymptotics of the Plancherel measure of the symmetric group and the limit form of Young tableaux. Dokl. Akad. Nauk SSSR 233, 1024–1027 (1977)
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Nagarajan, V., Sviridenko, M. (2008). Tight Bounds for Permutation Flow Shop Scheduling. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds) Integer Programming and Combinatorial Optimization. IPCO 2008. Lecture Notes in Computer Science, vol 5035. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68891-4_11
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