# Completing Partial Schedules for Open Shop with Unit Processing Times and Routing

• René van Bevern
• Artem V. Pyatkin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)

## Abstract

Open Shop is a classical scheduling problem: given a set $$\mathcal J$$ of jobs and a set $$\mathcal M$$ of machines, find a minimum-makespan schedule to process each job $$J_i\in \mathcal J$$ on each machine $$M_q\in \mathcal M$$ for a given amount $$p_{iq}$$ of time such that each machine processes only one job at a time and each job is processed by only one machine at a time. In Routing Open Shop, the jobs are located in the vertices of an edge-weighted graph $$\mathcal G=(V,E)$$ whose edge weights determine the time needed for the machines to travel between jobs. The travel times also have a natural interpretation as sequence-dependent family setup times. Routing Open Shop is NP-hard for $$|V|=|\mathcal M|=2$$. For the special case with unit processing times $$p_{iq}=1$$, we exploit Galvin’s theorem about list-coloring edges of bipartite graphs to prove a theorem that gives a sufficient condition for the completability of partial schedules. Exploiting this schedule completion theorem and integer linear programming, we show that Routing Open Shop with unit processing times is solvable in  time, that is, fixed-parameter tractable parameterized by $$|V|+|\mathcal M|$$. Various upper bounds shown using the schedule completion theorem suggest it to be likewise beneficial for the development of approximation algorithms.

## Keywords

NP-hard scheduling problem Fixed-parameter algorithm Edge list-coloring Sequence-dependent family or batch setup times

## References

1. 1.
Allahverdi, A., Ng, C.T., Cheng, T.C.E., Kovalyov, M.Y.: A survey of scheduling problems with setup times or costs. Eur. J. Oper. Res. 187(3), 985–1032 (2008)
2. 2.
Averbakh, I., Berman, O., Chernykh, I.: A $${\frac{6}{5}}$$-approximation algorithm for the two-machine routing open-shop problem on a two-node network. Eur. J. Oper. Res. 166(1), 3–24 (2005)
3. 3.
Averbakh, I., Berman, O., Chernykh, I.: The routing open-shop problem on a network: Complexity and approximation. Eur. J. Oper. Res. 173(2), 531–539 (2006)
4. 4.
Bellman, R.: Dynamic programming treatment of the Travelling Salesman Problem. J. ACM 9(1), 61–63 (1962)
5. 5.
van Bevern, R., Chen, J., Hüffner, F., Kratsch, S., Talmon, N., Woeginger, G.J.: Approximability and parameterized complexity of multicover by $$c$$-intervals. Inform. Process. Lett. 115(10), 744–749 (2015a)
6. 6.
van Bevern, R., Komusiewicz, C., Sorge, M.: Approximation algorithms for mixed, windy, and capacitated arc routing problems. In: Proceedings of the 15th ATMOS. OASIcs, vol. 48. Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2015b)Google Scholar
7. 7.
van Bevern, R., Mnich, M., Niedermeier, R., Weller, M.: Interval scheduling and colorful independent sets. J. Sched. 18, 449–469 (2015)
8. 8.
van Bevern, R., Niedermeier, R., Sorge, M., Weller, M.: Complexity of arc routing problems. In: Arc Routing: Problems, Methods, and Applications. SIAM (2014)Google Scholar
9. 9.
van Bevern, R., Niedermeier, R., Suchý, O.: A parameterized complexity view on non-preemptively scheduling interval-constrained jobs: few machines, small looseness, and small slack. J. Sched. (in press, 2016). doi:
10. 10.
Bodlaender, H.L., Fellows, M.R.: W[2]-hardness of precedence constrained $$k$$-processor scheduling. Oper. Res. Lett. 18(2), 93–97 (1995)
11. 11.
Chernykh, I., Kononov, A., Sevastyanov, S.: Efficient approximation algorithms for the routing open shop problem. Comput. Oper. Res. 40(3), 841–847 (2013)
12. 12.
Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Heidelberg (2015)
13. 13.
Dorn, F., Moser, H., Niedermeier, R., Weller, M.: Efficient algorithms for Eulerian Extension and Rural Postman. SIAM J. Discrete Math. 27(1), 75–94 (2013)
14. 14.
Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London (2013)
15. 15.
Fellows, M.R., McCartin, C.: On the parametric complexity of schedules to minimize tardy tasks. Theor. Comput. Sci. 298(2), 317–324 (2003)
16. 16.
Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2006)
17. 17.
Galvin, F.: The list chromatic index of a bipartite multigraph. J. Comb. Theory B 63(1), 153–158 (1995)
18. 18.
Gonzalez, T., Sahni, S.: Open shop scheduling to minimize finish time. J. ACM 23(4), 665–679 (1976)
19. 19.
Gutin, G., Jones, M., Sheng, B.: Parameterized complexity of the k-Arc Chinese postman problem. In: Schulz, A.S., Wagner, D. (eds.) ESA 2014. LNCS, vol. 8737, pp. 530–541. Springer, Heidelberg (2014)Google Scholar
20. 20.
Gutin, G., Jones, M., Wahlström, M.: Structural parameterizations of the mixed Chinese postman problem. In: Bansal, N., Finocchi, I. (eds.) ESA 2015. LNCS, vol. 9294, pp. 668–679. Springer, Heidelberg (2015)
21. 21.
Gutin, G., Muciaccia, G., Yeo, A.: Parameterized complexity of $$k$$-Chinese Postman Problem. Theor. Comput. Sci. 513, 124–128 (2013)
22. 22.
Gutin, G., Wahlström, M., Yeo, A.: Parameterized Rural Postman and Conjoining Bipartite Matching problems (2014). arXiv:1308.2599v4
23. 23.
Halldórsson, M.M., Karlsson, R.K.: Strip graphs: recognition and scheduling. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 137–146. Springer, Heidelberg (2006)
24. 24.
Held, M., Karp, R.M.: A dynamic programming approach to sequencing problems. J. SIAM 10(1), 196–210 (1962)
25. 25.
Hermelin, D., Kubitza, J.-M., Shabtay, D., Talmon, N., Woeginger, G.: Scheduling two competing agents when one agent has significantly fewer jobs. In: Proceedings of the 10th IPEC. LIPIcs, vol. 43, pp. 55–65. Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2015)Google Scholar
26. 26.
Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12(3), 415–440 (1987)
27. 27.
Kononov, A.: $$O(\log n)$$-approximation for the routing open shop problem. RAIRO-Oper. Res. 49(2), 383–391 (2015)
28. 28.
Lenstra, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)
29. 29.
Mnich, M., Wiese, A.: Scheduling and fixed-parameter tractability. Math. Program. 154(1–2), 533–562 (2015)
30. 30.
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)
31. 31.
Pyatkin, A.V., Chernykh, I.D.: The Open Shop problem with routing in a two-node network and allowed preemption (in Russian). Diskretnyj Analiz i Issledovaniye Operatsij 19(3), 65–78 (2012). English translation in J. Appl. Ind. Math. 6(3), 346–354
32. 32.
Serdyukov, A.I.: On some extremal by-passes in graphs (in Russian). Upravlyayemyye sistemy 17, 76–79 (1978). zbMATH 0475.90080
33. 33.
Slivnik, T.: Short proof of Galvin’s Theorem on the list-chromatic index of a bipartite multigraph. Comb. Probab. Comput. 5, 91–94 (1996)
34. 34.
Sorge, M., van Bevern, R., Niedermeier, R., Weller, M.: A new view on Rural Postman based on Eulerian Extension and Matching. J. Discrete Alg. 16, 12–33 (2012)
35. 35.
Yu, W., Liu, Z., Wang, L., Fan, T.: Routing open shop and flow shop scheduling problems. Eur. J. Oper. Res. 213(1), 24–36 (2011)
36. 36.
Zhu, X., Wilhelm, W.E.: Scheduling and lot sizing with sequence-dependent setup: A literature review. IIE Trans. 38(11), 987–1007 (2006)