In IWOCA 2019, Ruangwises and Itoh introduced stable noncrossing matchings, where participants of each side are aligned on each of two parallel lines, and no two matching edges are allowed to cross each other. They defined two stability notions, strongly stable noncrossing matching (SSNM) and weakly stable noncrossing matching (WSNM), depending on the strength of blocking pairs. They proved that a WSNM always exists and presented an \(O(n^{2})\)-time algorithm to find one for an instance with n men and n women. They also posed open questions of the complexities of determining existence of an SSNM and finding a largest WSNM. In this paper, we show that both problems are solvable in polynomial time. Our algorithms are applicable to extensions where preference lists may include ties, except for one case which we show to be NP-complete. This NP-completeness holds even if each person's preference list is of length at most two and ties appear in only men's preference lists. To complement this intractability, we show that the problem is solvable in polynomial time if the length of preference lists of one side is bounded by one (but that of the other side is unbounded).

在IWOCA 2019中，Ruangwises和Itoh引入了稳定的非交叉匹配，其中每一边的参与者在两条平行线上的每条线上对齐，并且不允许两个匹配边彼此交叉。他们根据阻塞对的强度定义了两个稳定概念，即强稳定的非交叉匹配（SSNM）和弱稳定的非交叉匹配（WSNM）。他们证明了WSNM始终存在，并提出了一种\（O（n ^ {2}）\） -time算法，该算法为n个男人和n个实例找到一个女性。他们还提出了确定SSNM存在和找到最大的WSNM的复杂性的公开问题。在本文中，我们证明了这两个问题在多项式时间内都是可以解决的。我们的算法适用于首选项列表可能包含联系的扩展名，但我们证明是NP完全的情况除外。即使每个人的偏好列表的长度最多为两个，并且领带仅出现在男性的偏好列表中，NP完整性也保持不变。为了补充这种难处理性，我们证明，如果一侧的偏好列表的长度由一侧限制（而另一侧的范围不受限制），则该问题可以在多项式时间内解决。