强稳定与极大弱稳定非交叉匹配
Strongly Stable and Maximum Weakly Stable Noncrossing Matchings
摘要：
在IWOCA 2019中，Ruangwises和Itoh引入了稳定的非交叉匹配，其中每一方的参与者都在两条平行线上对齐，并且不允许两个匹配的边缘相互交叉。他们定义了两个稳定性概念，强稳定非交叉匹配（SSNM）和弱稳定非交叉匹配（WSNM），这取决于阻塞对的强度。他们证明了一个WSNM总是存在的，并提出了一个$O(n^{2})$时间算法，以找到一个$n$男性和$n$女性的例子。他们还提出了确定SSNM存在和找到最大WSNM的复杂性的公开问题。在本文中，我们证明了这两个问题是可解的多项式时间。我们的算法是适用于扩展的偏好列表可能包括领带，除了一种情况下，我们证明是NP完全的。即使每个人的偏好列表的长度最多为2，并且领带只出现在男人的偏好列表中，这个NP完全性也成立。为了补充这个棘手的问题，我们证明了这个问题是可解的多项式时间，如果一侧的偏好列表的长度是有界的（但另一侧是无界的）。
英文摘要：
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).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
