Strong formulations for network design problems with connectivity requirements

The network design problem with connectivity requirements lNDCr includes as special cases a wide variety of celebrated combinatorial optimization problems including the minimum spanning tree, Steiner tree, and survivable network design problems. We develop strong formulations for two versions of the edge-connectivity NDC problem: unitary problems requiring connected network designs, and nonunitary problems permitting nonconnected networks as solutions. We l1r present a new directed formulation for the unitary NDC problem that is stronger than a natural undirected formulation; l2r project out two classes of valid inequalities—partition inequalities, and combinatorial design inequalities—that generalize known classes of valid inequalities for the Steiner tree problem to the unitary NDC problem; and l3r show how to strengthen and direct nonunitary problems. Our results provide a unifying framework for strengthening formulations for NDC problems, and demonstrate the power of flow-based formulations for network design problems with connectivity requirements. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45l2r, 61–79 2005

[1]  Thomas L. Magnanti,et al.  Connectivity–splitting models for survivable network design , 2004 .

[2]  Michel X. Goemans,et al.  A catalog of steiner tree formulations , 1993, Networks.

[3]  Ali Ridha Mahjoub,et al.  Two-edge connected spanning subgraphs and polyhedra , 1994, Math. Program..

[4]  M. R. Rao,et al.  The Steiner tree problem II: Properties and classes of facets , 1994, Math. Program..

[5]  Kamal Jain.  A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem , 2001, Comb..

[6]  Mohamed Didi Biha,et al.  K-edge Connected Polyhedra on Series-parallel Graphs , 1996, Oper. Res. Lett..

[7]  Martin Grötschel,et al.  Computational Results with a Cutting Plane Algorithm for Designing Communication Networks with Low-Connectivity Constraints , 1992, Oper. Res..

[8]  Mihalis Yannakakis,et al.  Expressing combinatorial optimization problems by linear programs , 1991, STOC '88.

[9]  C. Nash-Williams.  On Orientations, Connectivity and Odd-Vertex-Pairings in Finite Graphs , 1960, Canadian Journal of Mathematics.

[10]  M. R. Rao,et al.  The Steiner tree problem I: Formulations, compositions and extension of facets , 1994, Math. Program..

[11]  Michel X. Goemans,et al.  The Steiner tree polytope and related polyhedra , 1994, Math. Program..

[12]  Pawel Winter,et al.  Steiner problem in networks: A survey , 1987, Networks.

[13]  Michel X. Goemans,et al.  Analysis of Linear Programming Relaxations for a Class of Connectivity Problems , 1990 .

[14]  Mourad Baïou,et al.  Steiner 2-Edge Connected Subgraph Polytopes on Series-Parallel Graphs , 1997, SIAM J. Discret. Math..

[15]  Pawel Winter,et al.  Generalized Steiner Problem in Series-Parallel Networks , 1986, J. Algorithms.

[16]  T. C. Hu,et al.  Multi-Terminal Network Flows , 1961 .

[17]  H. Whitney.  Non-Separable and Planar Graphs. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[18]  Mechthild Stoer,et al.  Facets for Polyhedra Arising in the Design of Communication Networks with Low-Connectivity Constraints , 1992, SIAM J. Optim..

[19]  Clyde L. Monma,et al.  Computer-aided design procedures for survivable fiber optic networks , 1989, IEEE J. Sel. Areas Commun..

[20]  M. Stoer.  Design of Survivable Networks , 1993 .

[21]  Martin Grötschel,et al.  Polyhedral and Computational Investigations for Designing Communication Networks with High Survivability Requirements , 1995, Oper. Res..

[22]  Martin Grötschel,et al.  Integer Polyhedra Arising from Certain Network Design Problems with Connectivity Constraints , 1990, SIAM J. Discret. Math..

[23]  Michel X. Goemans,et al.  Worst-case comparison of valid inequalities for the TSP , 1995, Math. Program..

[24]  T. Magnanti,et al.  A Dual-Based Algorithm for Multi-Level Network Design , 1994 .

[25]  Egon Balas,et al.  The perfectly matchable subgraph polytope of a bipartite graph , 1983, Networks.

[26]  Richard T. Wong,et al.  A dual ascent approach for steiner tree problems on a directed graph , 1984, Math. Program..

[27]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1987, JACM.

[28]  Jacques F. Benders,et al.  Partitioning procedures for solving mixed-variables programming problems , 2005, Comput. Manag. Sci..

[29]  Mauro Dell'Amico,et al.  Annotated Bibliographies in Combinatorial Optimization , 1997 .

[30]  Sunil Chopra,et al.  Polyhedra of the Equivalent Subgraph Problem and Some Edge Connectivity Problems , 1992, SIAM J. Discret. Math..

[31]  K. Menger.  Zur allgemeinen Kurventheorie , 1927 .

[32]  S. Chopra.  On the spanning tree polyhedron , 1989 .

[33]  Sylvia C. Boyd,et al.  An Integer Polytope Related to the Design of Survivable Communication Networks , 1993, SIAM J. Discret. Math..

[34]  George L. Nemhauser,et al.  Handbooks in operations research and management science , 1989 .

[35]  S. Raghavan,et al.  Formulations and algorithms for network design problems with connectivity requirements , 1995 .