Skip to main content
SpringerLink
Log in
Book cover

Workshop on Algorithms and Data Structures

WADS 1997: Algorithms and Data Structures pp 307–320Cite as

Intractability of assembly sequencing: Unit disks in the plane

  • Session 10A: Invited Lecture
  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1272))

Abstract

We consider the problem of removing a given disk from a collection of unit disks in the plane. At each step, we allow a disk to be removed by a collision-free translation to infinity, and the goal is to access a given disk using as few steps as possible. This DISKS problem is a version of a common task in assembly sequencing, namely removing a given part from a fully assembled product. Recently there has been a focus on optimizing assembly sequences over various cost measures, however with very limited algorithmic success. We explain this lack of success, proving strong inapproximability results in this simple geometric setting. Namely, we show that approximating the number of steps required to within a factor of 2log1−γ n for any γ > 0 is quasi-NP-hard. This provides the first inapproximability results for assembly sequencing, realized in a geometric setting.

As a stepping stone, we study the approximability of scheduling with AND/OR precedence constraints. The DISKS problem can be formulated as a scheduling problem where the order of removals is to be scheduled. Before scheduling a disk to be removed, a path must be cleared, and so we get precedence constraints on the tasks; however, the form of such constraints differs from traditional scheduling in that there is a choice of which path to clear. We prove our main result by first showing the similar inapproximability of this scheduling problem, and then by showing that a sufficiently hard subproblem can be realized geometrically using unit disks in the plane. Furthermore, our construction is fairly robust, in that is can be placed on a polynomially sized integer grid, it remains valid even when we consider only horizontal and vertical translations, and it also applies to axis-aligned unit squares and higher dimensions.

Supported by ARO MURI Grant DAAH04-96-1-0007 and by NSF Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation.

Supported by an Alfred P. Sloan Research Fellowship, an IBM Faculty Partnership Award, an ARO MURI Grant DAAH04-96-1-0007, and NSF Young Investigator Award CCR-9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation.

This is a preview of subscription content, log in via an institution.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. Agarwal, M. de Berg, D. Halperin, and M. Sharir. Efficient generation of k-directional assembly sequences. In Proc. 7th ACM Symp. on Discrete Algorithms, pages 122–131, 1996.

    Google Scholar 

  2. S. Arora. Polynomial-time approximation schemes for Euclidean TSP and other geometric problems. In Proc. 37th Symp. on Found. Comput. Sci., pages 1–11, 1996.

    Google Scholar 

  3. S. Arora, L. Babai, J. Stern, and Z. Sweedyk. The hardness of approximate optima in lattices, codes and linear equations. In Proc. 34th Symp. on Found. Comput. Sci., pages 724–733, 1993.

    Google Scholar 

  4. S. Arora and C. Lund. Hardness of approximations. In D. Hochbaum, editor, Approximation Algorithms for NP-Hard Problems. PWS Publishing Company, Boston, MA, 1996.

    Google Scholar 

  5. B. Baker. Approximation algorithms for NP-complete problems on planar graphs. J. ACM, 41(1):153–180, 1994.

    Google Scholar 

  6. D. Baldwin. Algorithmic methods and software tools for the generation of mechanical assembly sequences. M.Sc. thesis, MIT, Cambridge, MA, 1990.

    Google Scholar 

  7. G. Boothroyd. Assembly Automation and Product Design. Marcel Dekker, Inc., New York, NY, 1991.

    Google Scholar 

  8. R. Dawson. On removing a ball without disturbing the others. Mathematics Magazine, 57(1):27–30, 1984.

    Google Scholar 

  9. M. de Berg, M. Overmars, and O. Schwarzkopf. Computing and verifying depth orders. SIAM J. Comput., 23(2):432–446, 1994.

    Google Scholar 

  10. F. Dehne and J.-R. Sack. Translation separability of polygons. Visual Computer, 3(4):227–235, 1987.

    Google Scholar 

  11. T. D. Fazio and D. Whitney. Simplified generation of all mechanical assembly sequences. IEEE Trans. on Robotics and Automation, 3(6):640–658, 1987.

    Google Scholar 

  12. U. Feige. A threshold of Inn for approximating set cover. In Proc. 28th ACM Symp. Theory Comput., pages 314–318, 1996.

    Google Scholar 

  13. M. Garey and D. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York, NY, 1979.

    Google Scholar 

  14. D. Gillies. Algorithms to schedule tasks with ANDIOR precedence constraints. Ph.D. thesis, University of Illinois, Urbana, IL, 1993.

    Google Scholar 

  15. D. Gillies and J. Liu. Scheduling tasks with AND/OR precedence constraints. SIAM J. Comput., 24(4):797–810, 1995.

    Google Scholar 

  16. M. Goldwasser. Complexity Measures for Assembly Sequencing. Ph.D. thesis, Dept. Comput. Sci., Stanford Univ., Stanford, CA, 1997.

    Google Scholar 

  17. M. Goldwasser, J.-C. Latombe, and R. Motwani. Complexity measures for assembly sequences. In Proc IEEE Int. Conf. on Robotics and Automation, pages 1581–1587, 1996.

    Google Scholar 

  18. S. Gottschlich, C. Ramos, and D. Lyons. Assembly and task planning: A taxonomy. IEEE Robotics and Automation Magazine, 1(3):4–12, 1994.

    Google Scholar 

  19. L. Guibas, D. Halperin, H. Hirukawa, and J.-C. L. R. Wilson. A simple and efficient procedure for polyhedral assembly partitioning under infinitesimal motions. In Proc IEEE Int. Conf. on Robotics and Automation, pages 2553–2560, 1995.

    Google Scholar 

  20. L. Guibas and F. Yao. On translating a set of rectangles. In F. Preparata, editor, Computational Geometry, Advances in Computing Research, pages 61–77. JAI Press Inc., 1983.

    Google Scholar 

  21. D. Halperin. Personal communication. 1995.

    Google Scholar 

  22. D. Halperin and R. Wilson. Assembly partitioning along simple paths: the case of multiple translations. In Proc. IEEE Int. Conf. on Robotics and Automation, pages 1585–1592, 1995.

    Google Scholar 

  23. J. Hstad. Clique is hard to approximate within n1−ε. In Proc. 37th Symp. on Found. Comput. Sci., pages 627–636, 1996.

    Google Scholar 

  24. R. Hoffman. A common sense approach to assembly sequence planning. In Computer-Aided Mechanical Assembly Planning, pages 289–314. Kluwer Academic Publishers, Boston, 1991.

    Google Scholar 

  25. L. Homem de Mello and A. Sanderson. Computer-Aided Mechanical Assembly Planning. Kluwer Academic Publishers, Boston, 1991.

    Google Scholar 

  26. L. Homem de Mello and A. Sanderson. A correct and complete algorithms for the generation of mechanical assembly sequences. IEEE Trans. on Robotics and Automation, 7(2):228–240, 1991.

    Google Scholar 

  27. J. Hopcroft, J. Schwartz, and M. Shaxir. On the complexity of motion planning for multiple independent objects: P-space hardness of the “Warehouseman's Problem”. Int. J. Robotics Research, 3(4):76–88, 1984.

    Google Scholar 

  28. D. Johnson. Approximation algorithms for combinatorial problems. J. Comput. Systems Sci., 9:256–278, 1974.

    Google Scholar 

  29. V. Kann. Polynomially bounded minimization problems which are hard to approximate. In Automata, Languages and Programming (Proc.,20th ICALP), volume 700 of Lecture Notes in Computer Science, pages 52–63. Springer-Verlag, 1993.

    Google Scholar 

  30. S. Kaufman, R. Wilson, R. Jones, T. Calton, and A. Ames. The Archimedes 2 mechanical assembly planning system. In Proc IEEE Int. Conf. on Robotics and Automation, pages 3361–3368, 1996.

    Google Scholar 

  31. L. Kavraki and M. Kolountzakis. Partitioning a planar assembly into two connected parts is NP-complete. Information Processing Letters, 55(3):159–165, 1995.

    Google Scholar 

  32. L. Kavraki, J.-C. Latombe, and R. Wilson. Complexity of partitioning an assembly. In Proc. 5th Canad. Conf. Comput. Geom., pages 12–17, Waterloo, Canada, 1993.

    Google Scholar 

  33. S. Khanna and R. Motwani. Towards a syntactic characterization of PTAS. In Proc. 28th ACM Symp. Theory Comput., pages 329–337, 1996.

    Google Scholar 

  34. S. Khanna, M. Sudan, and L. Trevisan. Constraint satisfaction: The approximability of minimization problems. In Proc. 12th IEEE Comput. Complexity Conference, 1997.

    Google Scholar 

  35. S. Lee and Y. Shin. Assembly planning based on geometric reasoning. Computers and Graphics, 14(2):237–250, 1990.

    Google Scholar 

  36. R. Motwani. Approximation algorithms. Stanford Technical Report STAN-CS-921435, 1992.

    Google Scholar 

  37. B. Natarajan. On planning assemblies. In Proc. 4th ACM Symp. on Computational Geometry, pages 299–308, 1988.

    Google Scholar 

  38. C. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes. J. Comput. Systems Sci., 43(3):425–440, 1991.

    Google Scholar 

  39. C. Papadimitriou and M. Yannakakis. The traveling salesman problem with distances one and two. Mathematics of Operations Research, 18(1):1–11, 1993.

    Google Scholar 

  40. B. Romney, C. Godard, M. Goldwasser, and G. Ramkumar. An efficient system for geometric assembly sequence generation and evaluation. In Proc. ASME Int. Computers in Engineering Conference, pages 699–712, 1995.

    Google Scholar 

  41. J. Snoeyink and J. Stolfi. Objects that cannot be taken apart with two hands. In Proc. ACM Symp. on Computational Geometry, pages 247–256, 1993.

    Google Scholar 

  42. G. Toussaint. Movable separability of sets. In G. Toussaint, editor, Computational Geometry, pages 335–375. North-Holland, Amsterdam, Netherlands, 1985.

    Google Scholar 

  43. R. Wilson. On Geometric Assembly Planning. Ph.D. thesis, Dept. Comput. Sci., Stanford Univ., Stanford, CA, 1992. Stanford Technical Report STAN-CS-92-1416.

    Google Scholar 

  44. R. Wilson, L. Kavraki, J.-C. Latombe, and T. Lozano-Pérez. Two-handed assembly sequencing. Int. J. of Robotics Research, 14(4):335–350, 1995.

    Google Scholar 

  45. R. Wilson and J.-C. Latombe. Geometric reasoning about mechanical assembly. Artificial Intelligence, 71(2):371–396, 1994.

    Google Scholar 

  46. J. Wolter. On the Automatic Generation of Plans for Mechanical Assembly. Ph.D. thesis, University of Michigan, 1988.

    Google Scholar 

  47. T. Woo and D. Dutta. Automatic disassembly and total ordering in three dimensions. J. Engineering for Industry, 113(2):207–213, 1991.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Stanford University, 94305-9045, Stanford, CA

    Michael Goldwasser & Rajeev Motwani

Authors
  1. Michael Goldwasser
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rajeev Motwani
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Frank Dehne Andrew Rau-Chaplin Jörg-Rüdiger Sack Roberto Tamassia

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Goldwasser, M., Motwani, R. (1997). Intractability of assembly sequencing: Unit disks in the plane. In: Dehne, F., Rau-Chaplin, A., Sack, JR., Tamassia, R. (eds) Algorithms and Data Structures. WADS 1997. Lecture Notes in Computer Science, vol 1272. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63307-3_70

Download citation

  • DOI: https://doi.org/10.1007/3-540-63307-3_70

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63307-5

  • Online ISBN: 978-3-540-69422-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation