A fast Lagrangian relaxation algorithm for finding multi-constrained multiple shortest paths

Finding a multi-constrained shortest path (MCSP) between a pair of nodes arises in many important applications such as quality of service provisioning in the next-generation network. While this problem subject to a single constraint has been well studied, efficient algorithms solving this problem with two or more constraints are still quite limited. In this paper, we propose a new Lagrangian relaxation algorithm for solving a generalized version of the MCSP problem, where we search for multiple shortest paths subject to multiple constraints. As in some related work, our algorithm first identifies the lower and upper bounds, and then tries to close the gap with a path enumeration procedure. However, our algorithm is based on the recognition that the Lagrange multipliers found by existing methods usually do not give the best search direction for minimizing path enumerations even though they can provide near-optimized lower bounds. We provide a solution to meet both of these goals. Through experiments on the most challenging benchmark instances, we show that our algorithm performs significantly better than the best known algorithm in the literature.