Skip to main content
SpringerLink
Log in

SDP Relaxations for Some Combinatorial Optimization Problems

  • Chapter
  • First Online:
Handbook on Semidefinite, Conic and Polynomial Optimization

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 166))

  • 5748 Accesses

Abstract

In this chapter we present recent developments on solving various combinatorial optimization problems by using semidefinite programming (SDP). We present several SDP relaxations of the quadratic assignment problem and the traveling salesman problem. Further, we show the equivalence of several known SDP relaxations of the graph equipartition problem, and present recent results on the bandwidth problem.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 219.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 279.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 279.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alizadeh, F.: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optimiz. 5, 13–51 (1995)

    Google Scholar 

  2. Anstreicher, K.M.: Recent advances in the solution of quadratic assignment problems. Math. Program. Ser. B 97, 27–42 (2003)

    Google Scholar 

  3. Anstreicher, K.M., Brixius, N.W.: A new bound for the quadratic assignment problem based on convex quadratic programming. Math. Program. Ser. A 89, 341–357 (2001)

    Google Scholar 

  4. Anstreicher, K.M., Brixius, N.W., Linderoth, J., Goux, J.-P.: Solving large quadratic assignment problems on computational grids. Math. Program. Ser. B 91, 563–588 (2002)

    Google Scholar 

  5. Anstreicher, K.M., Wolkowicz, H.: On Lagrangian relaxation of quadratic matrix constraints. SIAM J. Matrix Anal. App. 22(1), 41–55 (2000)

    Google Scholar 

  6. Biswas, R., Hendrickson, B., Karypis, G.: Graph partitioning and parallel computing. Parallel Comput. 26(12), 1515–1517 (2000)

    Google Scholar 

  7. Brixius, N.W., Anstreicher, K.M.: Solving quadratic assignment problems using convex quadratic programming relaxations. Optim. Method Softw. 16, 49–68 (2001)

    Google Scholar 

  8. Brixius, N.W., Anstreicher, K.M: The Steinberg wiring problem. In: Grötschel, M. (ed.) The Sharpest Cut: The Impact of Manfred Padberg and His work. SIAM, Philadelphia, USA (2004)

    Google Scholar 

  9. Burer, S., Vandenbussche, D.: Solving lift-and-project relaxations of binary integer programs. SIAM J. Optimiz. 16, 726–750 (2006)

    Google Scholar 

  10. Burkard, R., Karisch, S.E., Rendl, F.: QAPLIB – A quadratic assignment problem library. J. Global Optim. 10, 391–403 (1997)

    Google Scholar 

  11. Burkard, R., Cela, E., Pardalos, P.M., Pitsoulis, L.: The quadratic assignment problem. In: Du, D.-Z., Pardalos, P. M., (ed.) Handbook of Combinatorial Optimization, Kluwer, Dordrecht, The Netherlands 3, 241–337 (1999)

    Google Scholar 

  12. Burkard, R., Dell’Amico, M., Martello, S.: Assignment Problems. SIAM, Philadelphia, USA (2009)

    Google Scholar 

  13. Chinn, P.Z., Chvátalová, J., Dewdney, A.K., Gibbs, N.E.: The bandwidth problem for graphs and matrices – a survey. J. Graph Theory 6, 223–254 (1982)

    Google Scholar 

  14. Cvetković, D., Čangalović, M., Kovačević-Vujčić, V.: Semidefinite programming methods for the symmetric traveling salesman problem. In: Proceedings of the 7th International IPCO Conference, Springer-Verlag, London, UK, 126–136 (1999)

    Google Scholar 

  15. Çela, E.: The Quadratic Assignment Problem: Theory and Algorithms. Kluwer, Dordrecht, The Netherlands (1998)

    Google Scholar 

  16. Dai, W., Kuh, E.: Simultaneous floor planning and global routing for hierarchical building-block layout. IEEE T. Comput. Aided D. 5, 828–837 (1987)

    Google Scholar 

  17. Dantzig, G.B., Fulkerson, D.R., Johnson, S.M.: Solution of a large-scale traveling salesman problem. Oper. Res. 2, 393–410 (1954)

    Google Scholar 

  18. Díaz, J., Petit, J., Serna, M.: A survey on graph layout problems. ACM Comput. Surveys 34, 313–356 (2002)

    Google Scholar 

  19. De Klerk, E., Pasechnik, D.V., Sotirov, R.: On semidefinite programming relaxations of the traveling salesman problem. SIAM J. Optim. 19(4), 1559–1573 (2008)

    Google Scholar 

  20. De Klerk, E., Sotirov, R.: Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem. Math. Program. Ser. A 122(2), 225–246 (2010)

    Google Scholar 

  21. De Klerk, E., Sotirov, R.: Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry. Math. Program. Ser. A (to appear)

    Google Scholar 

  22. De Klerk, E., Dobre, C.: A comparison of lower bounds for symmetric circulant traveling salseman problem. Discrete. Appl. Math. (to appear)

    Google Scholar 

  23. De Klerk, E., Pasechnik, D.V., Sotirov, R., Dobre, C.: On semidefinite programming relaxations of maximum k-section. Math. Program. Ser. B (to appear)

    Google Scholar 

  24. Dobre C.: Semidefinite programming approaches for structured combinatorial optimization problems. PhD thesis, Tilburg University, The Netherlands, (2011)

    Google Scholar 

  25. Donath, W.E., Hoffman, A.J.: Lower bounds for the partitioning of graphs. IBM Journal of Res. Dev. 17, 420–425 (1973)

    Google Scholar 

  26. Ding, Y., Ge, D., Wolkowicz, H.: On Equivalence of Semidefinite Relaxations for Quadratic Matrix Programming. Math. Oper. Res., 36(1), 88–104 (2011)

    Google Scholar 

  27. Elshafei, A.N.: Hospital layout as a quadratic assignment problem. Oper. Res. Quart. 28, 167–179 (1977)

    Google Scholar 

  28. Eschermann, B., Wunderlich, H.J.: Optimized synthesis of self-testable finite state machines. In: 20th International Symposium on Fault-Tolerant Computing (FFTCS 20), Newcastle upon Tyne, UK (1990)

    Google Scholar 

  29. Fiduccia, C.M., Mattheyses, R.M.: A linear-time heuristic for improving network partitions. Proceedings of the 19th Design Automation Conference, 175–181 (1982)

    Google Scholar 

  30. Frank, M., Wolfe, P.: An algorithm for quadratic programming. Nav. Res. Log. Quart, 3, 95–110 (1956)

    Google Scholar 

  31. Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theoret. Comput. Sci. 1(3), 237–267 (1976)

    Google Scholar 

  32. Ghaddar, B., Anjos, M.F., Liers, F.: A branch-and-cut algorithm based on semidefinite programming for the minimum k–partition problem. Ann. Oper. Res. (2008). doi:10.1007/s10479-008-0481-4

    Article  Google Scholar 

  33. Gijswijt, D.: Matrix algebras and semidefinite programming techniques for codes. PhD thesis, University of Amsterdam, The Netherlands (2005)

    Google Scholar 

  34. Gilmore, P.C.: Optimal and suboptimal algorithms for the quadratic assignment problem. SIAM J. Appl. Math. 10, 305–31 (1962)

    Google Scholar 

  35. Goemans, M.X., Rendl, F.: Combinatorial optimization. In: Wolkowicz, H., Saigal, R., Vandenberghe, L. (ed.) Handbook of Semidefinite Programming: Theory, Algorithms and Applications, Kluwer (2000)

    Google Scholar 

  36. Haemers, W.H.: Interlacing eigenvalues and graphs. Linear Algebra Appl. 227/228, 593–616 (1995)

    Google Scholar 

  37. Harary, F.: Problem 16. In: Fiedler, M. (ed.) Theory of graphs and its applications, Czechoslovak Academy of Science, Prague (1967)

    Google Scholar 

  38. Harper, L.H.: Optimal assignments of numbers to vertices. J.  SIAM 12, 131–135 (1964)

    Google Scholar 

  39. Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees. Oper. Res. 18(6), 1138–1162 (1970)

    Google Scholar 

  40. Helmberg, C., Rendl, F., Mohar, B., Poljak, S.: A spectral approach to bandwidth and separator problems in graphs. Linear and Multilinear Algebra 39, 73–90 (1995)

    Google Scholar 

  41. Juvan, M., Mohar, B.: Laplace eigenvalues and bandwidth-type invariants of graphs. J. Graph Theory 17(3), 393–407 (1993)

    Google Scholar 

  42. Lai, Y.L., Williams, K.: A survey of solved problems and applications on bandwidth, edgesum and profiles of graphs. J. Graph Theory 31, 75–94 (1999)

    Google Scholar 

  43. Lengauer, T.: Combinaotrial Algorithms for Integrated Circuit Layout. Wiley, Chichester (1990)

    Google Scholar 

  44. Karisch, S.E.: Nonlinear approaches for quadratic assignment and graph partition problems. PhD thesis, Technical University Graz (1995)

    Google Scholar 

  45. Karisch, S.E., Rendl, F.: Semidefnite programming and graph equipartition. In: Topics in Semidefinite and Interior–Point Methods, The Fields Institute for research in Mathematical Sciences, Communications Series, vol. 18, pp. 77–95. Providence, Rhode Island (1998)

    Google Scholar 

  46. Kernighan, B.W., Lin, S.: An efficient heuristic procedure for partitioning graphs. Bell System Tech. J. 49, 291–307 (1970)

    Google Scholar 

  47. Krarup, J., Pruzan, P.M.: Computer-aided layout design. Math. Program. Stud. 9, 75–94 (1978)

    Google Scholar 

  48. Krarup, J.: Quadratic Assignment. Saertryk af data, 3–27 (1972)

    Google Scholar 

  49. Laporte, G., Mercure, H.: Balancing hydrulic turbine runners: a quadratic assignment problem. Eur. J. Oper. Res. 35, 378–382 (1988)

    Google Scholar 

  50. Lawler, E.: The quadratic assignment problem. Manage. Sci. 9, 586–599 (1963)

    Google Scholar 

  51. Lengauer, T.: Combinatorial Algorithms for Integrated Circuit layout. Wiley, Chicester (1990)

    Google Scholar 

  52. Lisser, A., Rendl, F.: Graph partitioning using linear and semidefinite programming. Math. Program. Ser. B 95(1), 91–101 (2003)

    Google Scholar 

  53. Lovász, L., Schrijver, A.: Cones of matrices and setfunctions and 01 optimization. SIAM J. Optimiz. 1(2), 166–190 (1991)

    Google Scholar 

  54. Mobasher, A., Taherzadeh, M., Sotirov, R., Khandani, A.K.: A near maximum likelihood decoding algorithm for MIMO systems based on semidefinite programming. IEEE Trans. on Info. Theory 53(11), 3869–3886 (2007)

    Google Scholar 

  55. Mobasher, A., Sotirov, R., Khandani, A.K.: Matrix-lifting SDP for detection in multiple antenna systems. IEEE Trans. on Signal Proc. 58(10), 5178–5185 (2010)

    Google Scholar 

  56. Nugent, C.E., Vollman, T.E., Ruml, J.: An experimental comparison of techniques for the assignment of facillities to locations. Oper. Res. 16, 150–173 (1968)

    Google Scholar 

  57. Nyström, M.: Solving certain large instances of the quadratic assignment problem: Steinberg’s examples. Technical report, California Institute of Technology, CA (1999)

    Google Scholar 

  58. Overton, M.L., Womersley, R.S.: On the sum of the largest eigenvalues of a symmetric matrix. SIAM J. Matrix Anal. Appl. 13, 41–45 (1992)

    Google Scholar 

  59. Papadimitriou, Ch.H.: The NP-Completeness of the bandwidth minimization problem. Computing 16(3), 263–270 (1976)

    Google Scholar 

  60. Povh, J., Rendl, F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Discrete Optim. 6(3), 231–241 (2009)

    Google Scholar 

  61. Povh, J., Rendl, F.: A copositive programming approach to graph partitioning. SIAM J. Optimiz. 18, 223–241 (2007)

    Google Scholar 

  62. Rendl, F., Sotirov, R.: Bounds for the quadratic assignment problem using the bundle method. Math. Program. Ser. B 109(2/3), 505–524 (2007)

    Google Scholar 

  63. Rendl, F., Wolkowicz, H.: A projection technique for partitioning nodes of a graph. Ann. Oper. Res. 58, 155–179 (1995)

    Google Scholar 

  64. Schrijver, A.: A comparison of the Delsarte and Lovász bounds. IEEE Trans. Inform. Theory 25, 425–429 (1979)

    Google Scholar 

  65. Steinberg, L.: The backboard wiring problem: a placement algorithm. SIAM Rev. 3, 37–50 (1961)

    Google Scholar 

  66. Zhao, Q., Karisch, S.E., Rendl, F., Wolkowicz, H.: Semidefinite programming relaxations for the quadratic assignment problem. J. Comb. Optim. 2, 71–109 (1998)

    Google Scholar 

  67. Wolkowicz, H., Zhao, Q.: Semidefinite programming relaxations for the graph partitioning. Discrete Appl. Math. 96/97, 461–479 (1999)

    Google Scholar 

Download references

Acknowledgements

The author would like to thank Edwin van Dam for valuable discussions and careful reading of this manuscript. The author would also like to thank an anonymous referee for suggestions that led to an improvement of this chapter.

Author information

Authors and Affiliations

  1. Department of Econometrics and Operations Research, Tilburg University, Warandelaan 2, 5000, LE, Tilburg, The Netherlands

    Renata Sotirov

Authors
  1. Renata Sotirov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Renata Sotirov .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Industrial, Ecole Polytechnique Montreal, Édouard-Montpetit Boulevard 2350, Montreal, QC, Canada, H3C 3A7

    Miguel F. Anjos

  2. LAAS-CNRS, Avenue du Colonel Roche 7, 31077, Toulouse Cedex 4, France

    Jean B. Lasserre

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Sotirov, R. (2012). SDP Relaxations for Some Combinatorial Optimization Problems. In: Anjos, M.F., Lasserre, J.B. (eds) Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science, vol 166. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0769-0_27

Download citation

Publish with us

Policies and ethics

Search

Navigation