We study the reconfiguration of odd matchings of combinatorial graphs. Odd matchings are matchings that cover all but one vertex of a graph. A reconfiguration step, or flip, is an operation that matches the isolated vertex and, consequently, isolates another vertex. The flip graph of odd matchings is a graph that has all odd matchings of a graph as vertices and an edge between two vertices if their corresponding matchings can be transformed into one another via a single flip. We show that computing the diameter of the flip graph of odd matchings is $Π_2^p$-hard. This complements a recent result by Wulf [FOCS25] that it is~$Π_2^p$-hard to compute the diameter of the flip graph of perfect matchings where a flip swaps matching edges along a single cycle of unbounded size. Further, we show that computing the radius of the flip graph of odd matchings is $Σ_3^p$-hard. The respective decision problems for the diameter and the radius are also complete in the respective level of the polynomial hierarchy. This shows that computing the radius of the flip graph of odd matchings is provably harder than computing its diameter, unless the polynomial hierarchy collapses. Finally, we reduce set cover to the problem of finding shortest flip sequences. As a consequence, we show $\log$-\APX-hardness and that the problem cannot be approximated by a sublogarithmic factor. By doing so, we answer a question asked by Aichholzer, Brenner, Dorfer, Hoang, Perz, Rieck, and Verciani [GD25].

翻译：本文研究组合图奇数匹配的重构问题。奇数匹配是指覆盖图中除一个顶点外所有顶点的匹配。重构步骤（或称翻转）是一种操作：将孤立顶点纳入匹配，同时使另一顶点变为孤立。奇数匹配的翻转图是以图的所有奇数匹配为顶点构成的图，若两个顶点对应的匹配能通过单次翻转相互转换，则其间存在边。我们证明计算奇数匹配翻转图的直径是$Π_2^p$-困难的。这补充了Wulf [FOCS25]的最新结果——计算完美匹配翻转图的直径是$Π_2^p$-困难的，其中翻转操作沿任意大尺寸单环交换匹配边。进一步，我们证明计算奇数匹配翻转图的半径是$Σ_3^p$-困难的。针对直径与半径的相应判定问题在多项式层次结构的对应层级上也是完备的。这表明除非多项式层次结构坍缩，否则计算奇数匹配翻转图的半径在可证明意义上比计算其直径更困难。最后，我们将集合覆盖问题归约至寻找最短翻转序列问题。由此证明该问题具有$\log$-\APX-难度，且无法获得亚对数因子的近似解。通过上述工作，我们回答了Aichholzer、Brenner、Dorfer、Hoang、Perz、Rieck和Verciani [GD25]提出的问题。