Finding shortest paths in mixed graphs is NP-hard even when the weighting is conservative. We give an FPT algorithm for this problem, where the parameter k is the number of "negative trees". We may always assume that the undirected edges have negative weight, and in a conservative weighting they form some trees -- these trees are called the negative trees. Our algorithm extends to mixed graphs with conservative weighting, where in each strongly connected component there are at most k negative trees. Necessarily we can also detect whether the weighting is conservative or not. If it is not conservative we give a strongly polynomial algorithm for the shortest exact (and bounded) walk problems, where we are looking the shortest walk from s to t consisting of exactly (at most) K edges.
Bibtex entry:
@techreport{egres-12-20,
AUTHOR | = | {Kir{\'a}ly, Zolt{\'a}n}, |
TITLE | = | {Shortest paths in mixed graphs}, |
NOTE | = | {{\tt egres.elte.hu}}, |
INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
YEAR | = | {2012}, |
NUMBER | = | {TR-2012-20} |