A Procedure for Computing the K Best Solutions to Discrete Optimization Problems and Its Application to the Shortest Path Problem

A general procedure is presented for computing the best, 2nd best,…, Kth best solutions to a given discrete optimization problem. If the number of computational steps required to find an optimal solution to a problem with n(0, 1) variables is c(n), then the amount of computation required to obtain the if best solutions is O(K_nc(n)).

The procedure specializes to published procedures of Murty and of Yen for the assignment problem and the shortest path problem, respectively. A method is presented for reducing the required amount of storage by a factor of n, compared with the algorithms of Murty and of Yen. It is shown how the K shortest (loopless) paths in an n-node network with positive and negative arcs can be computed with an amount of computation which is O(Kn³). This represents an improvement by a factor of n, compared with Yen's algorithm.