## neal young / Tarjan91Faster

• Networks 21(2):205-221(1991)
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 the fastest known 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 minimum-mean cycles (an important special case of both problems) is $$O(m + n \log n)$$.