Minimizing the weight of an edge set satisfying parity constraints is a challenging branch of combinatorial optimization as witnessed by the binary hypergraph chapter of Alexander Schrijver's book ``Combinatorial Optimization" (Chapter 80). This area contains relevant graph theory problems including open cases of the Max Cut problem and some multiflow problems. We clarify the interconnections between some of these problems and establish three levels of difficulties. On the one hand, we prove that the Shortest Odd Path problem in undirected graphs without cycles of negative total weight and several related problems are NP-hard, settling a long-standing open question asked by Lov\'asz (Open Problem 27 in Schrijver's book ``Combinatorial Optimization''). On the other hand, we provide an efficient algorithm to the closely related and well-studied Minimum-weight Odd $T$-Join problem for non-negative weights: our algorithm runs in FPT time parameterized by $c$, where $c$ is the number of connected components in some efficiently computed minimum-weight $T$-join. If negative weights are also allowed, then finding a minimum-weight odd $\{s,t\}$-join is equivalent to the Minimum-weight Odd $T$-Join problem for arbitrary weights, whose complexity is still only conjectured to be polynomial-time solvable. The analogous problems for digraphs are also considered.

翻译：满足奇偶约束的边集权重最小化问题是组合优化领域中的一个具有挑战性的分支，亚历山大·施赖弗在其著作《组合优化》第80章中通过二元超图章节对此进行了阐述。该领域包含相关图论问题，包括最大割问题的未解案例及若干多流问题。我们阐明了其中一些问题的相互关联，并建立了三个难度层级。一方面，我们证明了无负总权环的无向图中的最短奇路径问题及其若干相关问题为NP难问题，这解决了Lovász提出的长期未解问题（施赖弗著作《组合优化》中的开放问题27）。另一方面，我们为非负权重情况下密切相关且被广泛研究的最小权奇$T$-连接问题提供了高效算法：该算法以FPT时间运行，参数为$c$，其中$c$是某个高效计算的最小权$T$-连接中连通分量的数量。若允许负权重，则寻找最小权奇$\{s,t\}$-连接等价于任意权重下的最小权奇$T$-连接问题，而后者是否可在多项式时间内求解仍仅为猜想。本文还考虑了有向图中的类似问题。