Strongly Stable and Maximum Weakly Stable Noncrossing Matchings

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 $$O(n^{2})$$ 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).

Scores and clusters of Hungarian universities

We present an application preference, list-based framework to Hungarian universities, which allows different type of flexible aggregation, and hence, analysis and clustering of application data. A novel mathematical method is developed by which preference lists can be converted into scored rankings. The proposed approach is demonstrated in the case of Hungary covering the period of 2006–2015. Our method reveals that the efforts to leverage the geographical center–periphery differences did not fulfil the expectations of policy makers. Also, it turns out that a student's top preference is very difficult to influence, while recruiters may build their strategy on the information of the first but one choice.

Manipulating Gale-Shapley Algorithm: Preserving Stability and Remaining Inconspicuous

We study the problem of manipulation of the men-proposing Gale-Shapley algorithm by a single woman via permutation of her true preference list. Our contribution is threefold: First, we show that the matching induced by an optimal manipulation is stable with respect to the true preferences. Second, we identify a class of optimal manipulations called inconspicuous manipulations which, in addition to preserving stability, are also nearly identical to the true preference list of the manipulator (making the manipulation hard to be detected). Third, for optimal inconspicuous manipulations, we strengthen the stability result by showing that the entire stable lattice of the manipulated instance is contained inside the original lattice.​