Abstract
An integer linear program (ILP) is symmetric if its variables can be permuted without changing the structure of the problem. Areas where symmetric ILPs arise range from applied settings (scheduling on identical machines), to combinatorics (code construction), and to statistics (statistical designs construction). Relatively small symmetric ILPs are extremely difficult to solve using branch-and-cut codes oblivious to the symmetry in the problem. This paper reviews techniques developed to take advantage of the symmetry in an ILP during its solution. It also surveys related topics, such as symmetry detection, polyhedral studies of symmetric ILPs, and enumeration of all non isomorphic optimal solutions.
Supported by ONR grant N00014-03-1-0133.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F.A. Aloul, A. Ramani, I.L.Markov, and K.A. Sakallah, Solving difficult instances of Booleansatisfiability in the presence of symmetry, IEEE Transactions on CAD 22 (2003) 1117–1137.
K.M. Anstreicher, Recent advances in the solution of quadratic assignment problems, Mathematical Programming 97 (2003) 27–42.
D.L. Applegate, R.E. Bixby, V. Chvátal, and W.J. Cook, The Traveling Salesman Problem, A Computational Study, Princeton, 2006.
A. von Arnim, R. Schrader, and Y. Wang, The permutahedron of N-sparse posets, Mathematical Programming 75 (1996) 1–18.
L. Babai, E.M. Luks, and Á. Seress, Fast management of permutation groups I, SIAMJournal on Computing 26 (1997) 1310–1342.
E. Balas, A linear characterization of permutation vectors, Management Science Research Report 364, Carnegie Mellon University, Pittsburgh, PA, 1975.
C. Barnhart, E.L. Johnson, G.L. Nemhauser, M.W.P. Savelsbergh, and P.H. Vance, Branchand-price: Column generation for solving huge integer programs, Operations Research 46 (1998) 316–329.
M.S. Bazaraa and O. Kirca, A branch-and-bound based heuristic for solving the quadraticassignment problem, Naval Research Logistics Quarterly 30 (1983) 287–304.
R. Bertolo, P. Östergård, and W.D. Weakley, An updated table of binary/ternary mixed coveringcodes, Journal of Combinatorial Designs 12 (2004) 157–176.
W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The User Language, Journal of Symbolic Computations 24 (1997) 235–265.
C.A. Brown, L. Finkelstein, and P.W. Purdom, Backtrack searching in the presence of symmetry, Lecture Notes in Computer Science 357, Springer, 1989, pp. 99–110.
C.A. Brown, L. Finkelstein, and P.W. Purdom, Backtrack Searching in the Presence of Symmetry, Nordic Journal of Computing 3 (1996) 203–219.
C. Buchheim and M. J¨unger, Detecting symmetries by branch&cut, Mathematical Programming 98 (2003) 369–384.
C. Buchheim and M. Jünger, Linear optimization over permutation groups, Discrete Optimization 2 (2005) 308–319.
D.A. Bulutoglu and F. Margot, Classification of orthogonal arrays by integer programming, Journal of Statistical Planning and Inference 138 (2008) 654–666. 682 Franc¸ois Margot
G. Butler, Computing in permutation and matrix groups II: Backtrack algorithm, Mathematics of Computation 39 (1982) 671–680.
G. Butler, Fundamental Algorithms for Permutation Groups, Lecture Notes in Computer Science 559, Springer, 1991.
G. Butler and J.J. Cannon, Computing in permutation and matrix groups I: Normal closure,commutator subgroups, series, Mathematics of Computation 39 (1982) 663–670.
G. Butler and W.H. Lam, A general backtrack algorithm for the isomorphism problem ofcombinatorial objects, Journal of Symbolic Computation 1 (1985) 363–381.
P.J. Cameron, Permutation Groups, London Mathematical Society, Student Text 45, Cambridge University Press,1999.
M. Campêlo and R.C. Corrêa, A Lagrangian decomposition for the maximum stable set problem, Working Paper (2008), Universidade Federal do Cear´a, Brazil.
M. Campêlo, V.A. Campos, and R.C. Corrêa, Um algoritmo de Planos-de-Corte para onúumero cromático fracionário de um grafo, Pesquisa Operational 29 (2009) 179–193.
M. Campêlo, R. Corrêa, and Y. Frota, Cliques, holes and the vertex coloring polytope, Information Processing Letters 89 (2004) 159–164.
M. Campêlo, V. Campos, and R. Corrêa, On the asymmetric representatives formulation forthe vertex coloring problem, Electronic Notes in Discrete Mathematics 19 (2005) 337–343.
M. Campêlo, V. Campos, and R. Corrêa, On the asymmetric representatives formulation forthe vertex coloring problem, Discrete Applied Mathematics 156 (2008) 1097–1111.
R.D. Cameron, C.J. Colbourn, R.C. Read, and N.C.Wormald, Cataloguing the graphs on 10vertices, Journal of Graph Theory 9 (1985) 551–562.
T. Christof and G. Reinelt, Decomposition and parallelization techniques for enumeratingthe facets of combinatorial polytopes, International Journal on Computational Geometry and Applications 11 (2001) 423–437.
G. Cohen, I. Honkala, S. Litsyn, and A. Lobstein, Covering Codes, North Holland, 1997.
C.J. Colbourn and J.H. Dinitz (eds.): The CRC Handbook of Combinatorial Designs, CRC Press, 2007.
G. Cooperman, L. Finkelstein, and N. Sarawagi, A random base change algorithm for permutationgroups, Proceedings of the International Symposium on Symbolic and Algebraic Computations – ISSAC 90, ACM Press, 1990, pp. 161–168.
J. Crawford, M.L. Ginsberg, E. Luks, and A. Roy, Symmetry-breaking predicates for searchproblems, KR’96: Principles of Knowledge Representation and Reasoning (L.C. Aiello, J. Doyle, and S. Shapiro, eds.), 1996, pp. 148–159.
P. Darga, M.H. Liffiton, K.A. Sakallah, and I.L. Markov, Exploiting structure in symmetrygeneration for CNF, Proceedings of the 41st Design Automation Conference, San Diego 2004, pp. 530–534.
Z. Degraeve, W. Gochet, and R. Jans, Alternative formulations for a layout problem in thefashion industry, European Journal of Operational Research 143 (2002) 80–93.
M. Desrochers and F. Soumis, A column generation approach to the urban transit crewscheduling problem, Transportation Science 23 (1989) 1–13.
A. Deza, K. Fukuda, T. Mizutani, and C. Vo, On the face lattice of the metric polytope, Discrete and Computational Geometry: Japanese Conference (Tokyo, 2002), Lecture Notes in Computer Science 2866, Springer, 2003, pp. 118–128.
A. Deza, K. Fukuda, D. Pasechnik, and M. Sato, On the skeleton of the metric polytope, Discrete and Computational Geometry: Japanese Conference (Tokyo, 2000), in Lecture Notes in Computer Science 2098, Springer, 2001, pp. 125–136.
Y. Dumas, M. Desrochers, and F. Soumis, The pickup and delivery problem with time windows, European Journal of Operations Research 54 (1991) 7–22.
M. Elf, C. Gutwenger, M. J¨unger, and G. Rinaldi, Branch-and-cut algorithms for combinatorialoptimization and their implementation in ABACUS, in [59] (2001) 155–222.
T. Fahle, S. Shamberger, and M. Sellmann, Symmetry breaking, Proc. 7th International Conference on Principles and Practice of Constraint Programming – CP 2001, Lecture Notes in Computer Science 2239, Springer, 2001, pp. 93–107. 17 Symmetry in Integer Linear Programming 683
P. Flener, A. Frisch, B. Hnich, Z. Kiziltan, I. Miguel, J. Pearson, and T. Walsh, Symmetry inmatrix models, working paper APES-30-2001, 2001.
P. Flener, A. Frisch, B. Hnich, Z. Kiziltan, I. Miguel, J. Pearson, and T. Walsh, Breaking rowand column symmetries in matrix models, Proc. 8th International Conference on Principles and Practice of Constraint Programming – CP 2002, Lecture Notes in Computer Science 2470, Springer, 2002, pp. 462–476.
P. Flener, J. Pearson, M. Sellmann, P. van Hentenryck, and M. Ågren, Dynamic structural symmetry breaking for constraint satisfaction problems, DOI 10.1007/s10601-008-9059-7, Constraints 14 (2009).
F. Focacci andM.Milano, Global cut framework for removing symmetries, Proc. 7th International Conference on Principles and Practice of Constraint Programming – CP 2001, Lecture Notes in Computer Science 2239, Springer, 2001, pp. 77–92.
E.J. Friedman, Fundamental domains for integer programs with symmetries, Proceedings of COCOA 2007, Lecture Notes in Computer Science 4616, 2007, pp. 146–153.
I.P. Gent, W. Harvey, and T. Kelsey, Groups and constraints: Symmetry breaking duringsearch, Proc. 8th International Conference on Principles and Practice of Constraint Programming – CP 2002, Lecture Notes in Computer Science 2470, Springer, 2002, pp. 415–430.
I.P. Gent, W. Harvey, T. Kelsey, and S. Linton, Generic SBDD using computational grouptheory, Proc. 9th International Conference on Principles and Practice of Constraint Programming – CP 2003, Lecture Notes in Computer Science 2833, Springer, 2003, pp. 333–347.
I.P. Gent, T. Kelsey, S. Linton, I. McDonald, I. Miguel, and B. Smith, Conditional symmetrybreaking, Proc. 11th International Conference on Principles and Practice of Constraint Programming, Lecture Notes in Computer Science 3709, 2005, pp. 333–347.
I.P. Gent, T. Kelsey, S.T. Linton, J. Pearson, and C.M. Roney-Dougal, Groupoids and conditionalsymmetry, Proc. 13th International Conference on Principles and Practice of Constraint Programming, Lecture Notes in Computer Science 4741, 2007, pp. 823–830.
I.P. Gent, K.E. Petrie, and J.-F. Puget, Symmetry in constraint programming, Handbook of Constraint Programming (F. Rossi, P. van Beek, and T. Walsh eds.), Elsevier, 2006, pp. 329– 376.
I.P. Gent and B.M. Smith, Symmetry breaking in constraint programming, Proceedings of ECAI-2002, IOS Press, 2002, pp. 599–603.
D.M. Gordon and D.R. Stinson, Coverings, The CRC Handbook of Combinatorial Designs (C.J. Colbourn and J.H. Dinitz, eds.), CRC Press, 2007, pp. 365–372.
L.C. Grove and C.T. Benson, Finite Reflection Groups, Springer, 1985.
H. Hämäläinen, I. Honkala, S. Litsyn, and P. Östergård, Football pools–A game for mathematicians, American Mathematical Monthly 102 (1995) 579–588.
C.M. Hoffman, Group-Theoretic Algorithms and Graph Isomorphism, Lecture Notes in Computer Science 136, Springer, 1982.
D.F. Holt, B. Eick, and E.A. O’Brien, Handbook of Computational Group Theory, Chapman & Hall/CRC, 2004.
R. Jans, Solving lotsizing problems on parallel identical machines using symmetry breakingconstraints, INFORMS Journal on Computing 21 (2009) 123–136.
R. Jans and Z. Degraeve, A note on a symmetrical set covering problem: The lottery problem, European Journal of Operational Research 186 (2008) 104–110.
M. Jerrum, A compact representation for permutation groups, Journal of Algorithms 7 (1986) 60–78.
M. Jünger and D. Naddef (eds.), Computational Combinatorial Optimization, Lecture Notes in Computer Science 2241, Springer, 2001.
V. Kaibel, M. Peinhardt, and M.E. Pfetsch, Orbitopal fixing, Proceedings of the 12th International Integer Programming and Combinatorial Optimization Conference (M. Fischetti and D.P.Williamson, eds.), Lecture Notes in Computer Science 4513, Springer, 2007, pp. 74–88.
V. Kaibel and M.E. Pfetsch, Packing and partitioning orbitopes,Mathematical Programming 114 (2008) 1–36.
D.L. Kreher and D.R. Stinson, Combinatorial Algorithms, Generation, Enumeration, andSearch, CRC Press, 1999. 684 François Margot
E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, and D.B. Shmoys, The Traveling SalesmanProblem, Wiley, 1985.
J.S. Leon, On an algorithm for finding a base and a strong generating set for a group givenby generating permutations, Mathematics of Computation 35 (1980) 941–974.
J.S. Leon, Computing automorphism groups of combinatorial objects, Computational Group Theory (M.D. Atkinson, ed.), Academic Press, 1984, pp. 321–335.
L. Liberti, Automatic generation of symmetry-breaking constraints, Proceedings of COCOA 2008, Lecture Notes in Computer Science 5165, 2008, pp. 328–338.
J. Linderoth, F. Margot, and G. Thain, Improving bounds on the football pool problem viasymmetry: Reduction and high-throughput computing, to appear in INFORMS Journal on Computing, 2009.
C. Luetolf and F. Margot, A catalog of minimally nonideal matrices, Mathematical Methods of Operations Research 47 (1998) 221–241.
E. Luks, Permutation groups and polynomial-time computation, Groups and Computation (L. Finkelstein and W. Kantor, eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science 11 (1993) 139–175.
J.L. Marenco and P.A. Rey, The football pool polytope, Electronic Notes in Discrete Mathematics 30 (2008) 75–80.
F. Margot, Pruning by isomorphism in branch-and-cut, Mathematical Programming 94 (2002) 71–90.
F.Margot, Small covering designs by branch-and-cut,Mathematical Programming 94 (2003) 207–220.
F. Margot, Exploiting orbits in symmetric ILP, Mathematical Programming 98 (2003) 3–21.
F. Margot, Symmetric ILP: Coloring and small integers, Discrete Optimization 4 (2007) 40– 62.
T. Mautor and C. Roucairol, A new exact algorithm for the solution of quadratic assignmentproblems, Discrete Applied Mathematics 55 (1994) 281–293.
B.D. McKay, Nauty User’s Guide (Version 2.2), Computer Science Department, Australian National University, Canberra.
B.D. McKay, Isomorph-free exhaustive generation, Journal of Algorithms 26 (1998) 306– 324.
A. Mehrotra and M.A. Trick, A column generation approach for graph coloring, INFORMS Journal on Computing 8 (1996) 344–354.
A. Mehrotra and M.A. Trick, Cliques and clustering: A combinatorial approach, Operations Research Letters 22 (1998) 1–12.
I. M´endez-D`ıaz and P. Zabala, A branch-and-cut algorithm for graph coloring, Discrete Applied Mathematics 154 (2006) 826–847.
W.H. Mills and R.C. Mullin, Coverings and packings, Contemporary Design Theory: A Collection of Surveys (J.H. Dinitz and D.R. Stinson, eds.), Wiley, 1992, pp. 371–399.
G.L. Nemhauser and S. Park, A polyhedral approach to edge colouring, Operations Research Letters 10 (1991) 315–322.
G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization, Wiley, 1988.
P. Östergård and W. Blass, On the size of optimal binary codes of length 9 and coveringradius 1, IEEE Transactions on Information Theory 47 (2001) 2556–2557.
P. Östergård and A. Wassermann, A new lower bound for the football pool problem for sixmatches, Journal of Combinatorial Theory Ser. A 99 (2002) 175–179.
J. Ostrowski, Personal communication, 2007.
J. Ostrowski, J. Linderoth, F. Rossi, and S. Smirglio, Orbital branching, IPCO 2007: The Twelfth Conference on Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science 4513, Springer, 2007, pp. 104–118.
J. Ostrowski, J. Linderoth, F. Rossi, and S. Smirglio, Constraint orbital branching, IPCO 2008: The Thirteenth Conference on Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science 5035, Springer, 2008, pp. 225–239. 17 Symmetry in Integer Linear Programming 685
M.W. Padberg and G. Rinaldi, A branch-and-cut algorithm for the resolution of large scalesymmetric travelling salesman problems, SIAM Review 33 (1991) 60–100.
K.E. Petrie and B.M. Smith, Comparison of symmetry breaking methods in constraint programming, In Proceedings of SymCon05, 2005.
F. Plastria, Formulating logical implications in combinatorial optimisation, European Journal of Operational Research 140 (2002) 338–353.
J.-F. Puget, On the satisfiability of symmetrical constrainted satisfaction problems, Proceedings of the 7th International Symposium on Methodologies for Intelligent Systems, Lecture Notes in Artificial Intelligence 689, Springer, 1993, pp. 350–361.
J.-F. Puget, Symmetry breaking using stabilizers, Proc. 9th International Conference on Principles and Practice of Constraint Programming – CP 2003, Lecture Notes in Computer Science 2833, Springer, 2003, pp. 585–599.
J.-F. Puget, Automatic detection of variable and value symmetries, Proc. 11th International Conference on Principles and Practice of Constraint Programming – CP 2005, Lecture Notes in Computer Science 3709, Springer, 2005, pp. 475–489.
J.-F. Puget, Symmetry breaking revisited, Constraints 10 (2005) 23–46.
J.-F. Puget, A comparison of SBDS and dynamic lex constraints, In Proceeding of SymCon06, 2006, pp. 56–60.
A. Ramani, F.A. Aloul, I.L.Markov, and K.A. Sakallah, Breaking instance-independent symmetriesin exact graph coloring, Journal of Artificial Intelligence Research 26 (2006) 191– 224.
A. Ramani and I.L.Markov, Automatically exploiting symmetries in constraint programming, CSCLP 2004 (B. Faltings et al., eds.), Lecture Notes in Artificial Intelligence 3419, Springer, 2005, pp. 98–112.
R.C. Read, Every one a winner or how to avoid isomorphism search when cataloguing combinatorialconfigurations, Annals of Discrete Mathematics 2 (1978) 107–120.
P.A. Rey, Eliminating redundant solutions of some symmetric combinatorial integer programs, Electronic Notes in Discrete Mathematics 18 (2004) 201–206.
J.J. Rotman, An Introduction to the Theory of Groups, 4th edition, Springer, 1994.
D. Salvagnin, A Dominance Procedure for Integer Programming, Master’s thesis, University of Padova, 2005.
E. Seah and D.R. Stinson, An enumeration of non-isomorphic one-factorizations and Howelldesigns for the graph K 10minus a one-factor, Ars Combinatorica 21 (1986) 145–161.
Á. Seress, Nearly linear time algorithms for permutation groups: An interplay between theoryand practice, Acta Applicandae Mathematicae 52 (1998) 183–207.
Á. Seress, Permutation Group Algorithms, Cambridge Tracts in Mathematics 152, Cambridge University Press, 2003.
H.D. Sherali, B.M.P. Fraticelli, and R.D. Meller, Enhanced model formulations for optimalfacility layout, Operations Research 51 (2003) 629–644.
H.D. Sherali and J.C. Smith, Improving discrete model representations via symmetry considerations, Management Science 47 (2001) 1396–1407.
H.D. Sherali, J.C. Smith, and Y. Lee, Enhanced model representations for an intra-ring synchronousoptical network design problem allowing demand splitting, INFORMS Journal on Computing 12 (2000) 284–298.
B. Smith, Reducing symmetry in a combinatorial design problem, Proc. 8th International Conference on Principles and Practice of Constraint Programming – CP 2002, Lecture Notes in Computer Science 2470, Springer, 2002, pp. 207–213.
B.M. Smith, S.C. Brailsford, P.M. Hubbard, and H.P. Williams, The progressive party problem:Integer linear programming and constraint programming compared, Constraints 1 (1996) 119–138.
R.G. Stanton and J.A. Bates, A computer search for B-coverings, in Combinatorial Mathematics VII (R.W. Robinson, G.W. Southern, and W.D. Wallis, eds.), Lecture Notes in Computer Science 829, Springer, 1980, pp. 37–50.
The GAP Group: GAP – Groups, Algorithms, and Programming, Version 4.4.10 (2007), http://www.gap-system.org. 686 Franc¸ois Margot
P.H. Vance, Branch-and-price algorithms for the one-dimensional cutting stock problem, Computational Optimization and Applications 9 (1998) 111–228.
P.H. Vance, C. Barnhart, E.L. Johnson, and G.L. Nemhauser, Solving binary cutting stockproblems by column generation and branch-and-bound, Computational Optimization and Applications 3 (1994) 111–130.
P.H. Vance, C. Barnhart, E.L. Johnson, and G.L. Nemhauser, Airline crew scheduling: A newformulation and decomposition algorithm, Operations Research 45 (1997) 188–200.
L.A. Wolsey, Integer Programming, Wiley, 1998.
M. Yannakakis, Expressing combinatorial optimization problems by linear programs, Journal of Computer and System Sciences 43 (1991) 441–466.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Margot, F. (2010). Symmetry in Integer Linear Programming. In: Jünger, M., et al. 50 Years of Integer Programming 1958-2008. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68279-0_17
Download citation
DOI: https://doi.org/10.1007/978-3-540-68279-0_17
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-68274-5
Online ISBN: 978-3-540-68279-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)