Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T20:45:46.645Z Has data issue: false hasContentIssue false
  • English
  • Français

Branch and Cut based on the volume algorithm: Steiner trees in graphs and Max-cut

Published online by Cambridge University Press:  01 July 2006

Francisco Barahona  and
László Ladányi
Francisco Barahona
Affiliation:
IBM T. J. Watson research Center, Yorktown Heights, NY 10589, USA; [email protected]; [email protected]
László Ladányi
Affiliation:
IBM T. J. Watson research Center, Yorktown Heights, NY 10589, USA; [email protected]; [email protected]
Get access

Abstract

We present a Branch-and-Cut algorithm where the volume algorithm is applied instead of the traditionally used dual simplex algorithm to the linear programming relaxations in the root node of the search tree. This means that we use fast approximate solutions to these linear programs instead of exact but slower solutions. We present computational results with the Steiner tree and Max-Cut problems. We show evidence that one can solve these problems much faster with the volume algorithm based Branch-and-Cut code than with a dual simplex based one. We discuss when the volume based approach might be more efficient than the simplex based approach.

Type
Research Article
Information
RAIRO - Operations Research , Volume 40 , Issue 1 , January 2006 , pp. 53 - 73
Copyright
© EDP Sciences, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aneja, Y.P., An integer linear programming approach to the Steiner problem in graphs. Networks 20 (1980) 167–178. CrossRef
D. Applegate, R. Bixby, V. Chvátal and W. Cook, On the solution of traveling salesman problems, in Proc. of the International Congress of Mathematicians III (1998) 645–656.
Bahiense, L., Barahona, F. and Porto, O., Solving Steiner tree problems in graphs with Lagrangian relaxation. J. Comb. Optim. 7 (2003) 259–282. CrossRef
Bahiense, L., Maculan, N. and Sagastizábal, C., The volume algorithm revisited: relation with bundle methods. Math. Program. 94 (2002) 41–69. CrossRef
Barahona, F., Ground state magnetization of Ising spin glasses. Phys. Rev. B 49 (1994) 2864–2867. CrossRef
Barahona, F. and Anbil, R., The volume algorithm: producing primal solutions with a subgradient method. Math. Program. 87 (2000) 385–399. CrossRef
Barahona, F. and Anbil, R., On some difficult linear programs coming from set partitioning. Discrete Appl. Math. 118 (2002) 3–11. Third ALIO-EURO Meeting on Applied Combinatorial Optimization (Erice, 1999). CrossRef
F. Barahona, M. Grötschel, M. Jünger and G. Reinelt. An application of combinatorial optimization to statistical physics and circuit layout design, Oper. Res. 36 (1988) 493–513. CrossRef
Barahona, F., Jünger, M. and Reinelt, G., Experiments in quadratic 0-1 programming. Math. Program. 44 (1989) 127–137. CrossRef
Barahona, F. and Mahjoub, A., On the cut polytope. Math. Program. 36 (1986) 157–173. CrossRef
A. Caprara and M. Fischetti, Branch-and-cut algorithms, in Annotated Bibliographies in Combinatorial Optimization, edited by M.D. Amico, F. Maffioli and S. Martello, Wiley (1997) 45–63.
Chopra, S., Gorres, E.R. and Rao, M.R., Solving the Steiner tree problem in graphs using branch-and-cut. ORSA J. Comput. 4 (1992) 320–335. CrossRef
COIN-OR, http:www.coin-or.org.
De Simone, C., Diehl, M., Jünger, M., Mutzel, P., Reinelt, G. and Rinaldi, G., Exact ground states of Ising spin glasses: New experimental results with a branch and cut algorithm. J. Stat. Phys. 80 (1995) 487–496. CrossRef
C. De Simone, M. Diehl, M. Jünger, P. Mutzel, G. Reinelt and G. Rinaldi, Exact ground states of two-dimensional +–J Ising spin glasses. J. Stat. Phys. (1996).
De Simone, C. and Rinaldi, G., A cutting plane algorithm for the max-cut problem. Optim. Method. Softw. 3 (1994) 195–214. CrossRef
Edmonds, J., Optimum branchings. J. Res. Natl. Bur. Stand. 71B (1967) 233–240. CrossRef
Escudero, L.F., Guignard, M. and Malik, K., Lagrangean, A relax-and-cut approach for the sequential ordering problem with precedence relations. Ann. Oper. Res. 50 (1994) 219–237. CrossRef
J.J. Forrest, The COIN-OR Linear Program Solver (CLP), INFORMS Atlanta, (2003).
A. Frangioni, A. Lodi and G. Rinaldi, Optimizing over semimetric polytopes, in Integer programming and combinatorial optimization, Springer, Berlin Lect. Notes Comput. Sci. 3064 (2004) 431–443.
M.X. Goemans and Y.S. Myung, A catalog of Steiner tree formulations, Networks 23 (1993), pp. 19–28.
Guignard, M., Efficient cuts in lagrangean “Relax-and-Cut" schemes. Eur. J. Oper. Res. 105 (1998) 216–223. CrossRef
Held, M. and Karp, R.M., The travelling salesman problem and minimum spanning trees. Oper. Res. 18 (1970) 1138–1162. CrossRef
Held, M. and Karp, R.M., The travelling salesman problem and minimum spanning trees: Part II. Math. Program. 1 (1971) 6–25. CrossRef
Held, M., Wolfe, P. and Crowder, H.P., Validation of subgradient optimization. Math. Program. 6 (1974) 62–88. CrossRef
Helmberg, C. and Rendl, F., Solving quadratic (0,1)-problems by semidefinite programs and cutting planes. Math. Program. 82 (1998) 291–315.
T. Koch and A. Martin, Steinlib, http://elib.zib.de/steinlib/steinlib.php.
T. Koch and A. Martin, Solving Steiner tree problems in graphs to optimality, Networks 32 (1998) 207–232.
C. Lemaréchal, Nondifferentiable optimization, in Optimization, Hanbooks Oper. Res., edited by G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd, North Holland, 1989, pp. 529–572.
A. Lucena, Steiner problem in graphs: Lagrangian relaxation and cutting-planes, COAL Bull. 21 (1992) 2–7.
A. Lucena, Tight bounds for the Steiner problem in graphs, in Proc. Netflow 93 (1993) 147–154.
A. Lucena. and J. Beasley, A branch-and-cut algorithm for the Steiner problem in graphs. Networks 31 (1998), pp. 39–59.
Mitchell, J.E., Computational experience with an interior point cutting plane algorithm. SIAM J. Optim. 10 (2000) 1212–1227. CrossRef
Padberg, M. and Rinaldi, G., A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems. SIAM Rev. 33 (1991) 60–100. CrossRef
Polzin, T. and Daneshmand, S.V., Improved algorithms for the Steiner problem in networks. Discrete Appl. Math. 112 (2001) 263–300. Combinatorial Optimization Symposium (Brussels, 1998). CrossRef
H. Takahashi and A. Matsuyama, An approximated solution for the Steiner tree problem in graphs, Math. Japonica 254 (1980) 573–577.
E. Uchoa, M. Poggi de Aragão and C.C. Ribeiro, Preprocessing Steiner problems from VLSI layout, Networks 40 (2002) 38–50. CrossRef
Wolfe, P., A method of conjugate subgradients for minimizing nondifferentiable functions. Math. Program. Study 3 (1975) 145–173. CrossRef