39,273 research outputs found
Strong stability in the Hospitals/Residents problem
We study a version of the well-known Hospitals/Residents problem in which participants' preferences may involve ties or other forms of indifference. In this context, we investigate the concept of strong stability, arguing that this may be the most appropriate and desirable form of stability in many practical situations. When the indifference is in the form of ties, we describe an O(a^2) algorithm to find a strongly stable matching, if one exists, where a is the number of mutually acceptable resident-hospital pairs. We also show a lower bound in this case in terms of the complexity of determining whether a bipartite graph contains a perfect matching. By way of contrast, we prove that it becomes NP-complete to determine whether a strongly stable matching exists if the preferences are allowed to be arbitrary partial orders
Preference Elicitation in Matching Markets Via Interviews: A Study of Offline Benchmarks
The stable marriage problem and its extensions have been
extensively studied, with much of the work in the literature
assuming that agents fully know their own preferences over
alternatives. This assumption however is not always practical
(especially in large markets) and agents usually need
to go through some costly deliberation process in order to
learn their preferences. In this paper we assume that such
deliberations are carried out via interviews, where an interview
involves a man and a woman, each of whom learns
information about the other as a consequence. If everybody
interviews everyone else, then clearly agents can fully learn
their preferences. But interviews are costly, and we may
wish to minimize their use. It is often the case, especially
in practical settings, that due to correlation between agents’
preferences, it is unnecessary for all potential interviews to
be carried out in order to obtain a stable matching. Thus
the problem is to find a good strategy for interviews to be
carried out in order to minimize their use, whilst leading to a
stable matching. One way to evaluate the performance of an
interview strategy is to compare it against a na¨ıve algorithm
that conducts all interviews. We argue however that a more
meaningful comparison would be against an optimal offline
algorithm that has access to agents’ preference orderings under
complete information. We show that, unless P=NP, no
offline algorithm can compute the optimal interview strategy
in polynomial time. If we are additionally aiming for a
particular stable matching (perhaps one with certain desirable
properties), we provide restricted settings under which
efficient optimal offline algorithms exist
On the Multidimensional Stable Marriage Problem
We provide a problem definition of the stable marriage problem for a general
number of parties under a natural preference scheme in which each person
has simple lists for the other parties. We extend the notion of stability in a
natural way and present so called elemental and compound algorithms to generate
matchings for a problem instance. We demonstrate the stability of matchings
generated by both algorithms, as well as show that the former runs in
time.Comment: 8 page
Local search for stable marriage problems with ties and incomplete lists
The stable marriage problem has a wide variety of practical applications,
ranging from matching resident doctors to hospitals, to matching students to
schools, or more generally to any two-sided market. We consider a useful
variation of the stable marriage problem, where the men and women express their
preferences using a preference list with ties over a subset of the members of
the other sex. Matchings are permitted only with people who appear in these
preference lists. In this setting, we study the problem of finding a stable
matching that marries as many people as possible. Stability is an envy-free
notion: no man and woman who are not married to each other would both prefer
each other to their partners or to being single. This problem is NP-hard. We
tackle this problem using local search, exploiting properties of the problem to
reduce the size of the neighborhood and to make local moves efficiently.
Experimental results show that this approach is able to solve large problems,
quickly returning stable matchings of large and often optimal size.Comment: 12 pages, Proc. PRICAI 2010 (11th Pacific Rim International
Conference on Artificial Intelligence), Byoung-Tak Zhang and Mehmet A. Orgun
eds., Springer LNA
Stable marriage with general preferences
We propose a generalization of the classical stable marriage problem. In our
model, the preferences on one side of the partition are given in terms of
arbitrary binary relations, which need not be transitive nor acyclic. This
generalization is practically well-motivated, and as we show, encompasses the
well studied hard variant of stable marriage where preferences are allowed to
have ties and to be incomplete. As a result, we prove that deciding the
existence of a stable matching in our model is NP-complete. Complementing this
negative result we present a polynomial-time algorithm for the above decision
problem in a significant class of instances where the preferences are
asymmetric. We also present a linear programming formulation whose feasibility
fully characterizes the existence of stable matchings in this special case.
Finally, we use our model to study a long standing open problem regarding the
existence of cyclic 3D stable matchings. In particular, we prove that the
problem of deciding whether a fixed 2D perfect matching can be extended to a 3D
stable matching is NP-complete, showing this way that a natural attempt to
resolve the existence (or not) of 3D stable matchings is bound to fail.Comment: This is an extended version of a paper to appear at the The 7th
International Symposium on Algorithmic Game Theory (SAGT 2014
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
Local search for stable marriage problems
The stable marriage (SM) problem has a wide variety of practical
applications, ranging from matching resident doctors to hospitals, to matching
students to schools, or more generally to any two-sided market. In the
classical formulation, n men and n women express their preferences (via a
strict total order) over the members of the other sex. Solving a SM problem
means finding a stable marriage where stability is an envy-free notion: no man
and woman who are not married to each other would both prefer each other to
their partners or to being single. We consider both the classical stable
marriage problem and one of its useful variations (denoted SMTI) where the men
and women express their preferences in the form of an incomplete preference
list with ties over a subset of the members of the other sex. Matchings are
permitted only with people who appear in these lists, an we try to find a
stable matching that marries as many people as possible. Whilst the SM problem
is polynomial to solve, the SMTI problem is NP-hard. We propose to tackle both
problems via a local search approach, which exploits properties of the problems
to reduce the size of the neighborhood and to make local moves efficiently. We
evaluate empirically our algorithm for SM problems by measuring its runtime
behaviour and its ability to sample the lattice of all possible stable
marriages. We evaluate our algorithm for SMTI problems in terms of both its
runtime behaviour and its ability to find a maximum cardinality stable
marriage.For SM problems, the number of steps of our algorithm grows only as
O(nlog(n)), and that it samples very well the set of all stable marriages. It
is thus a fair and efficient approach to generate stable marriages.Furthermore,
our approach for SMTI problems is able to solve large problems, quickly
returning stable matchings of large and often optimal size despite the
NP-hardness of this problem.Comment: 12 pages, Proc. COMSOC 2010 (Third International Workshop on
Computational Social Choice
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