Abstract We study the problem of determining whether a given random matching can be implemented as a lottery over weakly stable deterministic matchings – a property known as ex-post stability. This concept arises in randomized allocation mechanisms such as school choice, where stability in each realized outcome is essential for fairness. Despite its importance in practice, the computational complexity of verifying ex-post stability has remained unresolved. We settle this question by showing that testing ex-post stability is NP-complete, even under highly restricted conditions – specifically, when both sides have dichotomous preferences or one of the sides has strict preferences. On the positive side, we present an integer programming formulation that finds a decomposition of a random matching with maximum weight on stable matchings. We also consider stronger versions of ex-post stability (in particular robust ex-post stability and ex-post strong stability) and prove that they can be tested in polynomial time.

We study fundamental connections between coalition formation games and clustering, illustrating the cross-disciplinary relevance of these concepts. We focus on graphical hedonic games where agents' preferences are compactly represented by a friendship graph and an enemy graph. In the context of clustering, friendship relations naturally align with data point similarities, whereas enmity corresponds to dissimilarities. We consider two stability notions based on single-agent deviations: local popularity and local stability. Exploring these concepts from an algorithmic viewpoint, we design efficient mechanisms for finding locally stable or locally popular partitions. Besides gaining theoretical insight into the computational complexity of these problems, we perform simulations that demonstrate how our algorithms can be successfully applied in clustering and community detection. Our findings highlight the interplay between coalition formation games and data-driven clustering techniques, offering fresh perspectives and applications in both areas.

In the Properly Colored Spanning Tree problem, we are given an edge-colored undirected graph and the goal is to find a properly colored spanning tree. The problem is interesting not only from a graph coloring point of view, but is also closely related to the Degree Bounded Spanning Tree and (1,2)-Traveling Salesman problems. We propose an optimization version called Maximum-size Properly Colored Forest problem, which aims to find a properly colored forest with as many edges as possible. We consider the problem in different graph classes and for different numbers of colors, and present polynomial-time approximation algorithms as well as inapproximability results for these settings. We also consider the Maximum-size Properly Colored Tree problem asking for the maximum size of a properly colored tree not necessarily spanning all the vertices. We show that the optimum is significantly more difficult to approximate than in the forest case, and provide an approximation algorithm for complete multigraphs.

Abstract. We consider the problem of finding a maximum popular matching in a many-to-many matching setting with two-sided preferences and matroid constraints. This problem was proposed by Kamiyama [ Theoret. Comput. Sci., 809 (2020), pp. 265–276] and solved in the special case where matroids are base orderable. Utilizing a newly shown matroid exchange property, we show that the problem is tractable for arbitrary matroids. We further investigate a different notion of popularity, where the agents vote with respect to lexicographic preferences, and show that both existence and verification problems become coNP-hard even in the [Formula: see text]-matching case.