Improve the complexity bound for parametric tropical eigenvector computation

Improve the worst-case time complexity for computing tropical eigenvectors of a parametric matrix from the current bound of O((m+n log n)n^3W) by effectively exploiting partial optimality of shortest paths.

Background

The paper extends an algorithm for the linear parametric minimum cycle mean problem to compute tropical eigenvalues and eigenvectors of parametric matrices. Although the eigenvalue computation has complexity O((m+n log n)n^2W), the stated worst-case complexity for computing the corresponding tropical eigenvectors is O((m+n log n)n^3W), owing in part to separately tracking shortest-path changes for different roots. The authors identify the possibility that partial optimality of shortest paths could be exploited to reduce this bound, but they do not establish such an improvement.