Strong formulations for network design problems with connectivity requirements

Authors

  • Thomas L. Magnanti,

    1. Sloan School of Management and Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
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  • S. Raghavan

    Corresponding author
    1. The Robert H. Smith School of Business and the Institute for Systems Research, Van Munching Hall, University of Maryland, College Park, Maryland 20742
    • The Robert H. Smith School of Business and the Institute for Systems Research, Van Munching Hall, University of Maryland, College Park, Maryland 20742
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Abstract

The network design problem with connectivity requirements (NDC) 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 (1) present a new directed formulation for the unitary NDC problem that is stronger than a natural undirected formulation; (2) 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 (3) 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. 45(2), 61–79 2005

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