Finding Multiple Shortest Paths Subject to Multiple Constraints: A Direction Tunning Algorithm

Numerous algorithms have been proposed for the well-known multi-constrained shortest path problem, but very few have good practical performance when there are two or more constraints. In this paper we propose a new Lagrangian relaxation algorithm that finds multiple shortest paths subject to multiple constraints. It follows the same procedure as some related work by first identifying the lower and upper bounds, and then closing the gap with a path enumeration algorithm. However, we bring up the recognition that the Lagrange multipliers found by existing methods, which although can provide a near-optimized lower bound, usually do not give the best search direction for path enumeration. Hence, we formulate the corresponding problem with a new objective function and provide an algorithm to find the multipliers meeting both goals. We then show how to incorporates feasibility checks with a state-of-the-art KSP algorithm to reduce enumerations. Experimental results on the most challenging benchmark instances indicate that, compared with the best known algorithm, not only can our algorithm solve a significantly larger number of instances to optimality, but its average running time for solved instances is much shorter, usually by one or two orders of magnitude.