Lagrangian relaxation and enumeration for solving constrained shortest‐path problems

The constrained shortest-path problem (CSPP) generalizes the standard shortest-path problem by adding one or more path-weight side constraints. We present a new algorithm for CSPP that Lagrangianizes those constraints, optimizes the resulting Lagrangian function, identifies a feasible solution, and then closes any optimality gap by enumerating near-shortest paths, measured with respect to the Lagrangianized length. “Near-shortest” implies -optimal, with a varying that equals the current optimality gap. The algorithm exploits a variety of techniques: a new path-enumeration method; aggregated constraints; preprocessing to eliminate edges that cannot form part of an optimal solution; “reprocessing” that reapplies preprocessing steps as improved solutions are found; and, when needed, a “phase-I procedure” to identify a feasible solution before searching for an optimal one. The new algorithm is often an order of magnitude faster than a state-of-the-art label-setting algorithm on singly constrained randomly generated grid networks. On multiconstrained grid networks, road networks, and networks for aircraft routing the advantage varies but, overall, the new algorithm is competitive with the labelsetting algorithm. © 2008 Wiley Periodicals, Inc.∗ NETWORKS, Vol. 00(0), 000–00