Faster Parametric Shortest Path and Minimum Balance Algorithms

TL;DR

This paper introduces faster algorithms for the parametric shortest path and minimum-balance problems, achieving improved theoretical runtime and demonstrating efficiency through empirical studies on random graphs.

Contribution

It presents new algorithms with O(nm + n^2 log n) complexity for both problems, advancing the computational methods for these graph problems.

Findings

Algorithms run in O(nm + n^2 log n) time.

Empirical results suggest expected time for minimum-mean cycle is O(n log n + m).

Demonstrates improved efficiency over previous methods.

Abstract

The parametric shortest path problem is to find the shortest paths in graph where the edge costs are of the form w_ij+lambda where each w_ij is constant and lambda is a parameter that varies. The problem is to find shortest path trees for every possible value of lambda. The minimum-balance problem is to find a ``weighting'' of the vertices so that adjusting the edge costs by the vertex weights yields a graph in which, for every cut, the minimum weight of any edge crossing the cut in one direction equals the minimum weight of any edge crossing the cut in the other direction. The paper presents fast algorithms for both problems. The algorithms run in O(nm+n^2 log n) time. The paper also describes empirical studies of the algorithms on random graphs, suggesting that the expected time for finding a minimum-mean cycle (an important special case of both problems) is O(n log(n) + m).