The Constrained Shortest Path Problem: Algorithmic Approaches and an Algebraic Study with Generalization

The constrained shortest path (CSP) problem requires the determination of a minimum cost s-t path with delay at most a nonzero integer T. In this paper, we first point out the equivalence of certain algorithms, simply called the LARAC (Lagrangian Relaxation Based Aggregated Cost) algorithm presented independently in some earlier works. The LARAC algorithm solves the integer relaxation of the CSP problem (RELAX-CSP) and is based on a geometric approach. We then present an algebraic study of RELAX-CSP and establish several new properties of the optimal solution. These properties also hold for general combinatorial optimization problems involving two additive parameters. We follow this by establishing a characterization of optimal solutions for the general CSP problem involving more than two additive parameters. We present a new heuristic called LARAC-BIN based on binary search. This heuristic involves a parameter whose value can be specified in advance depending on the allowable deviation of the cost from the optimum. Using Megiddo’s parametric search, we also present a strongly polynomial time algorithm for RELAX-CSP. This algorithm has the best complexity to date for RELAXCSP. Finally, we present an integrated approach to the CSP problem and show how the LARAC algorithm can be used to achieve considerable speedup of ε-approximation algorithms for the CSP problem.