Abstract
In this chapter we describe a novel way of deriving semidefinite programming relaxations of a wide class of combinatorial optimization problems. Many combinatorial optimization problems may be viewed as finding an induced subgraph of a specific type of maximum weight in a given weighted graph. The relaxations we describe are motivated by concepts from algebraic combinatorics. In particular, we consider a matrix algebra that contains the adjacency matrix of the required subgraph, and formulate a convex relaxation of this algebra. Depending on the type of subgraph, this algebra may be the Bose–Mesner algebra of an association scheme, or, more generally, a coherent algebra. Thus we obtain new (and known) relaxations of the traveling salesman problem, maximum equipartition problems in graphs, the maximum stable set problem, etc.
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References
Andersson, G.: An approximation algorithm for Max p-Section. Lecture Notes in Computer Science 1563, 236–247 (1999)
Bannai, E., Ito, T.: Algebraic combinatorics. I: Association schemes. The Benjamin/Cummings Publishing Co. Inc., Menlo Park, CA (1984)
Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-regular graphs. In: Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 18. Springer-Verlag, Berlin (1989)
Cvetković, D., Cangalović, M., Kovačević-Vujčić, V.: Semidefinite programming methods for the symmetric travelling salesman problem. In: Proceedings of the 7th International Conference on Integer Programming and Combinatorial Optimization, pp. 126–136. Springer-Verlag, London (1999)
Cameron, P.J.: Permutation Groups. London Mathematical Society Student Texts 45, Cambridge University Press, Cambridge (1999)
Feige U., Langberg, M.: Approximation algorithms for maximization problems arising in graph partitioning. Journal of Algorithms 41, 174–211 (2001)
Feige, U., Lee, J.R., Hajiaghayi, M.T.: Improved approximation algorithms for minimum-weight vertex separators. SIAM Journal on Computing 38(2), 629–657 (2008)
Frieze, A., Jerrum, M.: Improved approximation algorithms for MAX k-CUT and MAX BISECTION. Algorithmica 18, 67–81 (1997)
The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.4.12. http://www.gap-system.org (2008)
Godsil, C.D.: Algebraic Combinatorics. Chapman & Hall, New York (1993)
Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM Journal on Computing 24, 296–317 (1995)
Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM 42, 1115–1145 (1995)
Greub, W.: Multilinear Algebra. Springer-Verlag, New York (1978)
Han, Q., Ye, Y., Zhang, J.: An improved rounding method and semidefinite relaxation for graph partitioning. Mathematical Programming 92, 509–535 (2002)
Khot, S., Kindler, G., Mossel, E., O’Donnell, R.: Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?. SIAM Journal on Computing 37, 319–357 (2007)
de Klerk, E., Pasechnik, D.V., Sotirov, R.: On semidefinite programming relaxations of the travelling salesman problem. SIAM Journal on Optimization 19, 1559–1573 (2008)
de Klerk, E., Sotirov, R.: Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem. Mathematical Programming 122(2), 225–246 (2010)
de Klerk, E., Sotirov, R.: Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry. Mathematical Programming, to appear.
Laurent, M., Rendl, F.: Semidefinite programming and integer programming. In: Aardal, K., Nemhauser, G., Weismantel, R. (eds.) Handbook on Discrete Optimization, pp. 393–514. Elsevier, Amsterdam (2005)
Lovász, L.: On the Shannon capacity of a graph. IEEE Transactions on Information Theory 25, 1–7 (1979)
Manthey, B.: Minimum-weight cycle covers and their approximability. Discrete Applied Mathematics 157, 1470–1480 (2009)
Martin, W.J., Tanaka, H.: Commutative association schemes. European J. Combin.30(6), 1497–1525 (2009)
McEliece, R.J., Rodemich, E.R., Rumsey, Jr., H.C.: The Lovász bound and some generalizations. Journal of Combinatorics, Information & System Sciences, 134–152 (1978)
Pasechnik, D.V., Kini, K.: A GAP package for computation with coherent configurations. In Fukuda, K., van der Hoeven, J., Joswig, M., Takayama, N. (eds.) Mathematical Software - ICMS 2010, Third International Congress on Mathematical Software, Kobe, Japan, September 13–17, 2010. Proceedings. Lecture Notes in Computer Science, vol. 6327, pp. 69–72. Springer (2010)
Povh J., Rendl, F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Discrete Optimization 6, 231–241 (2009)
Schrijver, A.: A comparison of the Delsarte and Lovász bounds. IEEE Transactions on Information Theory 25, 425–429 (1979)
Schrijver, A.: New code upper bounds from the Terwilliger algebra. IEEE Transactions on Information Theory 51, 2859–2866 (2005)
Wedderburn, J.H.M.: On hypercomplex numbers. Proceedings of the London Mathematical Society 6, 77–118 (1907)
Ye, Y.: A.699 approximation algorithm for max-bisection. Mathematical Programming 90, 101–111 (2001)
Zhao, Q., Karisch, S.E., Rendl, F., Wolkowicz, H.: Semidefinite programming relaxations for the quadratic assignment problem. Journal of Combinatorial Optimization 2, 71-109 (1998)
Acknowledgements
Etienne de Klerk would like to thank Chris Godsil for many valuable discussions on the contents of this chapter.
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de Klerk, E., de Oliveira Filho, F.M., Pasechnik, D.V. (2012). Relaxations of Combinatorial Problems Via Association Schemes. 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_7
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