Stable Matchings with Covering Constraints: A Complete Computational Trichotomy Stable matching problems with lower quotas are fundamental in academic hiring and ensuring operability of rural hospitals. Only few tractable (polynomialtime solvable) cases of stable matching with lower quotas have been identified; most such problems are mathsf NPhard and also hard to approximate (Hamada et al. in Algorithmica 74(1):440xe2x80x93465, 2016). We therefore consider stable matching problems with lower quotas under a relaxed notion of tractability, namely fixedparameter tractability. By cloning hospitals we focus on the case when all hospitals have upper quota equal toxc2xa01, which generalizes the setting of xe2x80x9carranged marriagesxe2x80x9d first considered by Knuth (Mariages stables et leurs relations avec dxe2x80x99autres problemes combinatoires, Les Presses de lxe2x80x99Universite de Montreal, Montreal, 1976). We investigate how a set of natural parameters, namely the maximum length of preference lists for men and women, the number of distinguished men and women, and the number of blocking pairs allowed determine the computational tractability of this problem. Our main result is a complete complexity trichotomy: for each choice of parameters we either provide a polynomialtime algorithm, or an mathsf NPhardness proof and fixedparameter algorithm, or mathsf NPhardness proof and mathsf W1hardness proof. As corollary, we negatively answer a question by Hamada et al. (Algorithmica 74(1):440xe2x80x93465, 2016) by showing fixedparameter intractability parameterized by optimal solution size. We also classify all cases of onesided constraints where only women may be distinguished.

An Algorithm for Strong Stability in the StudentProject Allocation Problem with Ties We study a variant of the StudentProject Allocation problem with lecturer preferences over Students where ties are allowed in the preference lists of students and lecturers (spast). We investigate the concept of strong stability in this context. Informally, a matching is strongly stable if there is no student and lecturer l such that if they decide to form a private arrangement outside of the matching via one of lxe2x80x99s proposed projects, then neither party would be worse off and at least one of them would strictly improve. We describe the first polynomialtime algorithm to find a strongly stable matching or report that no such matching exists, given an instance of spast. Our algorithm runs in (O(m2)) time, where m is the total length of the studentsxe2x80x99 preference lists.