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In IWOCA 2019, Ruangwises and Itoh introduced stable noncrossing matchings,where participants of each side are aligned on each of two parallel lines, andno two matching edges are allowed to cross each other. They defined twostability notions, strongly stable noncrossing matching (SSNM) and weaklystable noncrossing matching (WSNM), depending on the strength of blockingpairs. They proved that a WSNM always exists and presented an $O(n^{2})$-timealgorithm to find one for an instance with $n$ men and $n$ women. They alsoposed open questions of the complexities of determining existence of an SSNMand finding a largest WSNM. In this paper, we show that both problems aresolvable in polynomial time. Our algorithms are applicable to extensions wherepreference lists may include ties, except for one case which we show to beNP-complete. This NP-completeness holds even if each person's preference listis of length at most two and ties appear in only men's preference lists. Tocomplement this intractability, we show that the problem is solvable inpolynomial time if the length of preference lists of one side is bounded by one(but that of the other side is unbounded).

Strongly Stable and Maximum Weakly Stable Noncrossing Matchings