Faster parametric shortest path and minimum-balance algorithms

We use Fibonacci heaps to improve a parametric shortest path algorithm of Karp and Orlin, and we combine our algorithm and the method of Schneider and Schneider’s minimum-balance algorithm to obtain a faster minimum-balance algorithm. For a graph with n vertices and m edges, our parametric shortest path algorithm and our minimum-balance algorithm both run in O(nm + n logn) time, improved from O(nm logn) for the parametric shortest path algorithm of Karp and Orlin and O(nm) for the minimum-balance algorithm of Schneider and Schneider. An important application of the parametric shortest path algorithm is in finding a minimum mean cycle. Experiments on random graphs suggest that the expected time for finding a minimum mean cycle with our algorithm is O(n logn+m).