29,731 research outputs found
A constraint programming approach to the hospitals/residents problem
An instance I of the Hospitals/Residents problem (HR) involves a set of residents (graduating medical students) and a set of hospitals, where each hospital has a given capacity. The residents have preferences for the hospitals, as do hospitals for residents. A solution of I is a <i>stable matching</i>, which is an assignment of residents to hospitals that respects the capacity conditions and preference lists in a precise way. In this paper we present constraint encodings for HR that give rise to important structural properties. We also present a computational study using both randomly-generated and real-world instances. We provide additional motivation for our models by indicating how side constraints can be added easily in order to solve hard variants of HR
Finding large stable matchings
When ties and incomplete preference lists are permitted in the stable marriage and hospitals/residents problems, stable matchings can have different sizes. The problem of finding a maximum cardinality stable matching in this context is known to be NP-hard, even under very severe restrictions on the number, size, and position of ties. In this article, we present two new heuristics for finding large stable matchings in variants of these problems in which ties are on one side only. We describe an empirical study involving these heuristics and the best existing approximation algorithm for this problem. Our results indicate that all three of these algorithms perform significantly better than naive tie-breaking algorithms when applied to real-world and randomly-generated data sets and that one of the new heuristics fares slightly better than the other algorithms, in most cases. This study, and these particular problem variants, are motivated by important applications in large-scale centralized 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
The Stable Roommates problem with short lists
We consider two variants of the classical Stable Roommates problem with Incomplete (but strictly ordered) preference lists (sri) that are degree constrained, i.e., preference lists are of bounded length. The first variant, egald-sri, involves finding an egalitarian stable matching in solvable instances of sri with preference lists of length at most d. We show that this problem is NP-hard even if d = 3. On the positive side we give a 2d+372d+37-approximation algorithm for d ∈{3,4,5} which improves on the known bound of 2 for the unbounded preference list case. In the second variant of sri, called d-srti, preference lists can include ties and are of length at most d. We show that the problem of deciding whether an instance of d-srti admits a stable matching is NP-complete even if d = 3. We also consider the “most stable” version of this problem and prove a strong inapproximability bound for the d = 3 case. However for d = 2 we show that the latter problem can be solved in polynomial time
Locally Stable Marriage with Strict Preferences
We study stable matching problems with locality of information and control.
In our model, each agent is a node in a fixed network and strives to be matched
to another agent. An agent has a complete preference list over all other agents
it can be matched with. Agents can match arbitrarily, and they learn about
possible partners dynamically based on their current neighborhood. We consider
convergence of dynamics to locally stable matchings -- states that are stable
with respect to their imposed information structure in the network. In the
two-sided case of stable marriage in which existence is guaranteed, we show
that the existence of a path to stability becomes NP-hard to decide. This holds
even when the network exists only among one partition of agents. In contrast,
if one partition has no network and agents remember a previous match every
round, a path to stability is guaranteed and random dynamics converge with
probability 1. We characterize this positive result in various ways. For
instance, it holds for random memory and for cache memory with the most recent
partner, but not for cache memory with the best partner. Also, it is crucial
which partition of the agents has memory. Finally, we present results for
centralized computation of locally stable matchings, i.e., computing maximum
locally stable matchings in the two-sided case and deciding existence in the
roommates case.Comment: Conference version in ICALP 2013; to appear in SIAM J. Disc Mat
Efficient algorithms for optimal matching problems under preferences
In this thesis we consider efficient algorithms for matching problems involving preferences,
i.e., problems where agents may be required to list other agents that they find
acceptable in order of preference. In particular we mainly study the Stable Marriage
problem (SM), the Hospitals / Residents problem (HR) and the Student / Project Allocation
problem (SPA), and some of their variants. In some of these problems the aim
is to find a stable matching which is one that admits no blocking pair. A blocking pair
with respect to a matching is a pair of agents that prefer to be matched to each other
than their assigned partners in the matching if any.
We present an Integer Programming (IP) model for the Hospitals / Residents problem
with Ties (HRT) and use it to find a maximum cardinality stable matching. We also
present results from an empirical evaluation of our model which show it to be scalable
with respect to real-world HRT instance sizes.
Motivated by the observation that not all blocking pairs that exist in theory will lead
to a matching being undermined in practice, we investigate a relaxed stability criterion
called social stability where only pairs of agents with a social relationship have the
ability to undermine a matching. This stability concept is studied in instances of
the Stable Marriage problem with Incomplete lists (smi) and in instances of hr. We
show that, in the smi and hr contexts, socially stable matchings can be of varying
sizes and the problem of finding a maximum socially stable matching (max smiss and
max hrss respectively) is NP-hard though approximable within 3/2. Furthermore we
give polynomial time algorithms for three special cases of the problem arising from
restrictions on the social network graph and the lengths of agents’ preference lists.
We also consider other optimality criteria with respect to social stability and establish
inapproximability bounds for the problems of finding an egalitarian, minimum regret
and sex equal socially stable matching in the sm context.
We extend our study of social stability by considering other variants and restrictions
of max smiss and max hrss. We present NP-hardness results for max smiss even
under certain restrictions on the degree and structure of the social network graph as
well as the presence of master lists. Other NP-hardness results presented relate to the
problem of determining whether a given man-woman pair belongs to a socially stable
matching and the problem of determining whether a given man (or woman) is part of
at least one socially stable matching. We also consider the Stable Roommates problem
with Incomplete lists under Social Stability (a non-bipartite generalisation of smi under
social stability). We observe that the problem of finding a maximum socially stable
matching in this context is also NP-hard. We present efficient algorithms for three
special cases of the problem arising from restrictions on the social network graph and
the lengths of agents’ preference lists. These are the cases where (i) there exists a
constant number of acquainted pairs (ii) or a constant number of unacquainted pairs
or (iii) each preference list is of length at most 2.
We also present algorithmic results for finding matchings in the spa context that are
optimal with respect to profile, which is the vector whose ith component is the number
of students assigned to their ith-choice project. We present an efficient algorithm for
finding a greedy maximum matching in the spa context — this is a maximum matching
whose profile is lexicographically maximum. We then show how to adapt this algorithm
to find a generous maximum matching — this is a matching whose reverse profile is
lexicographically minimum. We demonstrate how this approach can allow additional
constraints, such as lecturer lower quotas, to be handled flexibly. We also present
results of empirical evaluations carried out on both real world and randomly generated
datasets. These results demonstrate the scalability of our algorithms as well as some
interesting properties of these profile-based optimality criteria.
Practical applications of spa motivate the investigation of certain special cases of the
problem. For instance, it is often desired that the workload on lecturers is evenly distributed
(i.e. load balanced). We enforce this by either adding lower quota constraints
on the lecturers (which leads to the potential for infeasible problem instances) or adding
a load balancing optimisation criterion. We present efficient algorithms in both cases.
Another consideration is the fact that certain projects may require a minimum number
of students to become viable. This can be handled by enforcing lower quota constraints
on the projects (which also leads to the possibility of infeasible problem instances). A
technique of handling this infeasibility is the idea of closing projects that do not meet
their lower quotas (i.e. leaving such project completely unassigned). We show that the
problem of finding a maximum matching subject to project lower quotas where projects
can be closed is NP-hard even under severe restrictions on preference lists lengths and
project upper and lower quotas. To offset this hardness, we present polynomial time
heuristics that find large feasible matchings in practice. We also present ip models
for the spa variants discussed and show results obtained from an empirical evaluation
carried out on both real and randomly generated datasets. These results show that
our algorithms and heuristics are scalable and provide good matchings with respect to
profile-based optimalit
Matching Dynamics with Constraints
We study uncoordinated matching markets with additional local constraints
that capture, e.g., restricted information, visibility, or externalities in
markets. Each agent is a node in a fixed matching network and strives to be
matched to another agent. Each agent has a complete preference list over all
other agents it can be matched with. However, depending on the constraints and
the current state of the game, not all possible partners are available for
matching at all times. For correlated preferences, we propose and study a
general class of hedonic coalition formation games that we call coalition
formation games with constraints. This class includes and extends many recently
studied variants of stable matching, such as locally stable matching, socially
stable matching, or friendship matching. Perhaps surprisingly, we show that all
these variants are encompassed in a class of "consistent" instances that always
allow a polynomial improvement sequence to a stable state. In addition, we show
that for consistent instances there always exists a polynomial sequence to
every reachable state. Our characterization is tight in the sense that we
provide exponential lower bounds when each of the requirements for consistency
is violated. We also analyze matching with uncorrelated preferences, where we
obtain a larger variety of results. While socially stable matching always
allows a polynomial sequence to a stable state, for other classes different
additional assumptions are sufficient to guarantee the same results. For the
problem of reaching a given stable state, we show NP-hardness in almost all
considered classes of matching games.Comment: Conference Version in WINE 201
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