Shortest Odd Paths in Conservative Graphs: Connections and Complexity

We present some reductions between optimization problems for undirected conservative graphs, that is, edge-weighted graphs without negative cycles. We settle the complexity of some of them, and exhibit some remaining challenges. Our key result is that the shortest odd path problem between two given vertices, and its variants, such as the shortest odd cycle problem through a given vertex, turn out to be NP-hard, deciding a long-standing question by Lov\'asz (Open Problem 27 in Schrijver's book, 2003), in the negative. The complexity of finding a shortest odd cycle for conservative weights or of finding an odd $T$-join of minimum cardinality remains open. We finally relate these problems to relevant, solved or hopeful variants.