Jointly Stable Matchings

In the stable marriage problem, we are given a set of men, a set of women, and each person\u27s preference list. Our task is to find a stable matching, that is, a matching admitting no unmatched (man, woman)-pair each of which improves the situation by being matched together. It is known that any instance admits at least one stable matching. In this paper, we consider a natural extension where k (>= 2) sets of preference lists L_i (1 <= i <= k) over the same set of people are given, and the aim is to find a jointly stable matching, a matching that is stable with respect to all L_i. We show that the decision problem is NP-complete already for k=2, even if each person\u27s preference list is of length at most four, while it is solvable in linear time for any k if each man\u27s preference list is of length at most two (women\u27s lists can be of unbounded length). We also show that if each woman\u27s preference lists are same in all L_i, then the problem can be solved in linear time

Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties

Motivated by the serious problem that hospitals in rural areas suffer from a shortage of residents, we study the Hospitals/Residents model in which hospitals are associated with lower quotas and the objective is to satisfy them as much as possible. When preference lists are strict, the number of residents assigned to each hospital is the same in any stable matching because of the well-known rural hospitals theorem; thus there is no room for algorithmic interventions. However, when ties are introduced to preference lists, this will no longer apply because the number of residents may vary over stable matchings. In this paper, we formulate an optimization problem to find a stable matching with the maximum total satisfaction ratio for lower quotas. We first investigate how the total satisfaction ratio varies over choices of stable matchings in four natural scenarios and provide the exact values of these maximum gaps. Subsequently, we propose a strategy-proof approximation algorithm for our problem; in one scenario it solves the problem optimally, and in the other three scenarios, which are NP-hard, it yields a better approximation factor than that of a naive tie-breaking method. Finally, we show inapproximability results for the above-mentioned three NP-hard scenarios

An improved approximation lower bound for finding almost stable maximum matchings

In the stable marriage problem that allows incomplete preference lists, all stable matchings for a given instance have the same size. However, if we ignore the stability, there can be larger matchings. Biró et al. defined the problem of finding a maximum cardinality matching that contains minimum number of blocking pairs. They proved that this problem is not approximable within some constant δ>1unless P=NP, even when all preference lists are of length at most 3. In this paper, we improve this constant δ to n(1−ε) for any ε>0, where n is the number of men in an input

Hard variants of stable marriage

The Stable Marriage Problem and its many variants have been widely studied in the literature (Gusfield and Irving, The Stable Marriage Problem: Structure and Algorithms, MIT Press, Cambridge, MA, 1989; Roth and Sotomayor, Two-sided matching: a study in game-theoretic modeling and analysis, Econometric Society Monographs, vol. 18, Cambridge University Press, Cambridge, 1990; Knuth, Stable Marriage and its Relation to Other Combinatorial Problems, CRM Proceedings and Lecture Notes, vol. 10, American Mathematical Society, Providence, RI, 1997), partly because of the inherent appeal of the problem, partly because of the elegance of the associated structures and algorithms, and partly because of important practical applications, such as the National Resident Matching Program (Roth, J. Political Economy 92(6) (1984) 991) and similar large-scale matching schemes. Here, we present the first comprehensive study of variants of the problem in which the preference lists of the participants are not necessarily complete and not necessarily totally ordered. We show that, under surprisingly restrictive assumptions, a number of these variants are hard, and hard to approximate. The key observation is that, in contrast to the case where preference lists are complete or strictly ordered (or both), a given problem instance may admit stable matchings of different sizes. In this setting, examples of problems that are hard are: finding a stable matching of maximum or minimum size, determining whether a given pair is stable––even if the indifference takes the form of ties on one side only, the ties are at the tails of lists, there is at most one tie per list, and each tie is of length 2; and finding, or approximating, both an `egalitarian' and a `minimum regret' stable matching. However, we give a 2-approximation algorithm for the problems of finding a stable matching of maximum or minimum size. We also discuss the significant implications of our results for practical matching schemes

Effect of Stoichiometry on Shape Memory Properties and Functional Stability of Ti–Ni–Pd Alloys

Ti⁻Ni⁻Pd shape memory alloys are promising candidates for high-temperature actuators operating at above 373 K. One of the key issues in developing high-temperature shape memory alloys is the degradation of shape memory properties and dimensional stabilities because plastic deformation becomes more pronounced at higher working temperature ranges. In this study, the effect of the Ti:(Ni + Pd) atomic ratio in TixNi70-xPd30 alloys with Ti content in the range from 49 at.% to 52 at.% on the martensitic transformation temperatures, microstructures and shape memory properties during thermal cycling under constant stresses were investigated. The martensitic transformation temperatures decreased with increasing or decreasing Ti content from the stoichiometric composition. In both Ti-rich and Ti-lean alloys, the transformation temperatures decreased during thermal cycling and the degree of decrease in the transformation temperatures became more pronounced as the composition of the alloy departed from the stoichiometric composition. Ti₂Pd and P phases were formed during thermal cycling in Ti-rich and Ti-lean alloys, respectively. Both Ti-rich and Ti-lean alloys exhibited superior dimensional stabilities and excellent shape memory properties with higher recovery ratio and larger work output during thermal cycling under constant stresses when compared with the alloys with near-stoichiometric composition