Shortest paths in mixed graphs and nearly conservative digraphs

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 weights, 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. We also give a generalization for directed graphs, where any negative cycle has length 2.