Stable Matchings with Covering Constraints: A Complete Computational Trichotomy

Abstract

Stable matching problems with lower quotas are fundamental in academic hiring and ensuring operability of rural hospitals. Only few tractable (polynomial-time solvable) cases of stable matching with lower quotas have been identified; most such problems are NP-hard and also hard to approximate (Hamada et al. in Algorithmica 74(1):440-465, 2016). We therefore consider stable matching problems with lower quotas under a relaxed notion of tractability, namely fixed-parameter tractability. By cloning hospitals we focus on the case when all hospitals have upper quota equal to 1, which generalizes the setting of "arranged marriages" first considered by Knuth (Mariages stables et leurs relations avec d'autres problèmes combinatoires, Les Presses de l'Université de Montréal, 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 polynomial-time algorithm, or an NP-hardness proof and fixed-parameter algorithm, or NP-hardness proof and W[1]-hardness proof. As corollary, we negatively answer a question by Hamada et al. (Algorithmica 74(1):440-465, 2016) by showing fixed-parameter intractability parameterized by optimal solution size. We also classify all cases of one-sided constraints where only women may be distinguished.

Envy-freeness and Relaxed Stability for Lower-Quotas : A Parameterized Perspective

“…They show that both these problems are NP-hard and cannot be approximated within 21 19 − ϵ, ϵ > 0 unless P = NP. These hardness results hold when the quotas are at most 1 and under other severe restrictions.…”

“…In the two-sided preference model, Parameterized Complexity is well-studied [18,17,9,2,19]. In [17] and [2], they study the well-known NP-hard problem -computing a maximum size stable matching in the stable marriage setting (a special case of HR setting, where the two sets represent men and women and the upper-quotas are 1) where the preference lists are incomplete and contain ties.…”

“…In [9], they explore the stable marriage setting with the goal of matching more agents at the expense of allowing minimum number of blocking pairs and present parameterized results for several parameters. In [19], they explore the problem studied in [11] from parameterized perspective and consider several parameters namely -the length of preference lists, number of agents with lower-quota, the number of blocking pairs allowed and the solution size. Organization of the paper.…”

Stable roommates with narcissistic, single-peaked, and single-crossing preferences

“…First, to better understand the NPcompleteness result, one may study the parameterized complexity with respect to the "degree" of incompleteness of the input preferences, such as the number of ties or the number of agents that are in the same equivalence class of the tie-relation. We refer to some recent papers on the parameterized complexity of preference-based stable matching problems [1,11,12,[14][15][16]29,[40][41][42]45] for this line of research. Second, we were not able to settle the computational complexity for complete preferences that are also single-peaked and single-crossing.…”

Multi-parameter Coordinated Public School Admission Model by using Stable Marriage

“…In its origin, this algorithm is illustrated in the matching process between equal number of men and women [10]. In practical, this algorithm is used in the matching process between colleges and students [12] and between hospitals and residents [13]. This algorithm also has been implemented in New York school admission system [14].…”