Abstract
We study the polyhedron P(G) defined by the convex hull of 2-edge-connected subgraphs of G where multiple copies of edges may be chosen. We show that each vertex of P(G) is also a vertex of the LP relaxation. Given the close relationship with the Graphical Traveling Salesman problem (GTSP), we examine how polyhedral results for GTSP can be modified and applied to P(G). We characterize graphs for which P(G) is integral and study how this relates to a similar result for GTSP. In addition, we show how one can modify some classes of valid inequalities for GTSP and produce new valid inequalities and facets for P(G).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Applegate, R. Bixby, V. Chvátal, and W. Cook, “Concorde: A code for solving traveling salesman problems,” http://www.tsp.gatech.edu/concorde.html.
F. Barahona and A.R. Mahjoub, “On two-connected subgraph polytopes,” Discrete Math., vol. 147, pp. 19–34, 1995.
S.C. Boyd and W.H. Cunningham, “Small traveling salesman polytopes,” Math. Oper. Res., vol. 16, no. 2, pp. 259–271, 1991.
S.C. Boyd and T. Hao, “An integer polytope related to the design of survivable communication networks,” SIAM J. Discrete Math., vol. 6, no. 4, pp. 612–630, 1993.
G. Chaty and M. Chein, “Minimally 2-edge-connected graphs,” J. Graph Theory, vol. 3, pp. 15–22, 1979.
S. Chopra, “The k-edge-connected spanning subgraph polyhedron,” SIAM J. Discrete Math., vol. 7, no. 2, pp. 245–259, 1994.
T. Christof and A. Lóbel, “PORTA: A polyhedron representation transformation algorithm,” http://www.zib.de/Optimization/Software/Porta/, 1997.
G. Cornuéjols, J. Fonlupt, and D. Naddef, “The travelling salesman problem on a graph and some related integer polyhedra,” Math. Prog., vol. 33, pp. 1–27, 1985.
J. Fonlupt and A.R. Mahjoub, “Critical extreme points of the 2-edge connected spanning subgraph polytope,” in Integer Programming and Combinatorial Optimization, volume 1610 of lecture notes in Comput. Sci., Springer, Berlin, 1999, pp. 166–182.
J. Fonlupt and D. Naddef, “The travelling salesman problem in graphs with some excluded minors,” Math. Prog., vol. 53, no. 2, pp. 147–172, 1992.
M. Grötschel and M.W. Padberg, “On the symmetric travelling salesman problem I: Inequalities,” Math. Prog., vol. 16, no. 3, pp. 265–280, 1979.
M. Grötschel and W.R. Pulleyblank, “Clique tree inequalities and the symmetric travelling salesman problem,” Math. Oper. Res., vol. 11, no. 4, pp. 537–569, 1986.
M. Grótschel, C.L. Monma, and M. Stoer, “Design of survivable networks,” in Network Models, volume 7 of Handbooks Oper. Res. Management Sci., North-Holland, Amsterdam, 1995, pp. 617–672.
A.R. Mahjoub. “Two-edge connected spanning subgraphs and polyhedra,” Math. Prog., vol. 64, no. 2, pp. 199–208, 1994.
A.R. Mahjoub, “On perfectly two-edge connected graphs,” Discrete Math., vol. 170, no. 1–3, pp. 153–172, 1997.
D. Naddef, “The binested inequalities for the symmetric travelling salesman polytope,” Math. Oper. Res., vol. 17, no. 4, pp. 882–900, 1992.
D. Naddef and G. Rinaldi, “The graphical relaxation: A new framework for the symmetric travelling salesman polytope,” Math. Prog., vol. 58, no. 1, pp. 53–88, 1992.
D. Naddef and S. Thienel, “Efficient separation routines for the symmetric traveling salesman problem. I. general tools and comb separation,” Math. Program., vol. 92 no. 2, Ser. A, pp. 237–255, 2002a.
D. Naddef and S. Thienel, “Efficient separation routines for the symmetric traveling salesman problem. II. Separating multi handle inequalities,” Math. Program., vol. 92, no. 2, Ser. A, 2002b.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Vandenbussche, D., Nemhauser, G.L. The 2-Edge-Connected Subgraph Polyhedron. J Comb Optim 9, 357–379 (2005). https://doi.org/10.1007/s10878-005-1777-9
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10878-005-1777-9