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, or some multiflow problems. We clarify the interconnections of some problems and establish three levels of difficulties. On the one hand, we prove that the Shortest Odd Path problem in an undirected graph 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 a polynomial-time algorithm to the closely related and well-studied Minimum-weight Odd $\{s,t\}$-Join problem for non-negative weights, whose complexity, however, was not known; more generally, we solve the Minimum-weight Odd $T$-Join problem in FPT time when parameterized by $|T|$. 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 only conjectured to be polynomially 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是组合优化中一个具有挑战性的分支，正如Alexander Schrijver的著作《组合优化》第80章关于二元超图的内容所证实。该领域包含相关的图论问题，例如最大割问题的未解情形或多流问题。我们厘清了若干问题之间的相互关联，并建立了三个难度层次。一方面，我们证明了无负总权重环的无向图中的最短奇数路径问题及若干相关问题是NP难的，解决了Lovász提出的长期开放问题（Schrijver《组合优化》中的开放问题27）。另一方面，我们为权重非负且紧密相关的、已被广泛研究的带权最小奇数$\{s,t\}$-连接问题提供了多项式时间算法——此前其复杂度未知；更一般地，当参数化为$|T|$时，我们在固定参数可解时间内解决了带权最小奇数$T$-连接问题。若允许负权重，则寻找最小权重奇数$\{s,t\}$-连接等价于任意权重下的最小权重奇数$T$-连接问题，其复杂度仅被猜想为多项式可解。本文还考虑了对偶图（有向图）中的类似问题。