311 research outputs found
Socially stable matchings in the hospitals / residents problem
In the Hospitals/Residents (HR) problem, agents are partitioned into hospitals and residents. Each agent wishes to be matched to an agent in the other set and has a strict preference over these potential matches. A matching is stable if there are no blocking pairs, i.e., no pair of agents that prefer each other to their assigned matches. Such a situation is undesirable as it could lead to a deviation in which the blocking pair form a private arrangement outside the matching. This however assumes that the blocking pair have social ties or communication channels to facilitate the deviation. Relaxing the stability definition to take account of the potential lack of social ties between agents can yield larger stable matchings.
In this paper, we define the Hospitals/Residents problem under Social Stability (HRSS) which takes into account social ties between agents by introducing a social network graph to the HR problem. Edges in the social network graph correspond to resident-hospital pairs in the HR instance that know one another. Pairs that do not have corresponding edges in the social network graph can belong to a matching M but they can never block M. Relative to a relaxed stability definition for HRSS, called social stability, we show that socially stable matchings can have different sizes and the problem of finding a maximum socially stable matching is NP-hard, though approximable within 3/2. Furthermore we give polynomial time algorithms for three special cases of the problem
Popular Matchings with Lower Quotas
We consider the well-studied Hospital Residents (HR) problem in the presence of lower quotas (LQ). The input instance consists of a bipartite graph G = (R U H, E) where R and H denote sets of residents and hospitals, respectively. Every vertex has a preference list that imposes a strict ordering on its neighbors. In addition, each hospital has an associated upper-quota and a lower-quota. A matching M in G is an assignment of residents to hospitals, and M is said to be feasible if every resident is assigned to at most one hospital and a hospital is assigned at least its lower-quota many residents
and at most its upper-quota many residents.
Stability is a de-facto notion of optimality in a model where both sets of vertices have preferences. A matching is stable if no unassigned pair has an incentive to deviate from it. It is well-known that an instance of the HRLQ problem need not admit a feasible stable matching. In this paper, we consider the notion of popularity for the HRLQ problem. A matching M is popular if no other matching M\u27 gets more votes than M when vertices vote between M and M\u27. When there are no lower quotas, there always exists a stable matching and it is known that every stable matching is popular.
We show that in an HRLQ instance, although a feasible stable matching need not exist, there is always a matching that is popular in the set of feasible matchings. We give an efficient algorithm to compute a maximum cardinality matching that is popular amongst all the feasible matchings in an HRLQ instance
Stable marriage and roommates problems with restricted edges: complexity and approximability
In the Stable Marriage and Roommates problems, a set of agents is given, each of them having a strictly ordered preference list over some or all of the other agents. A matching is a set of disjoint pairs of mutually acceptable agents. If any two agents mutually prefer each other to their partner, then they block the matching, otherwise, the matching is said to be stable. We investigate the complexity of finding a solution satisfying additional constraints on restricted pairs of agents. Restricted pairs can be either forced or forbidden. A stable solution must contain all of the forced pairs, while it must contain none of the forbidden pairs.
Dias et al. (2003) gave a polynomial-time algorithm to decide whether such a solution exists in the presence of restricted edges. If the answer is no, one might look for a solution close to optimal. Since optimality in this context means that the matching is stable and satisfies all constraints on restricted pairs, there are two ways of relaxing the constraints by permitting a solution to: (1) be blocked by as few as possible pairs, or (2) violate as few as possible constraints n restricted pairs.
Our main theorems prove that for the (bipartite) Stable Marriage problem, case (1) leads to View the MathML source-hardness and inapproximability results, whilst case (2) can be solved in polynomial time. For non-bipartite Stable Roommates instances, case (2) yields an View the MathML source-hard but (under some cardinality assumptions) 2-approximable problem. In the case of View the MathML source-hard problems, we also discuss polynomially solvable special cases, arising from restrictions on the lengths of the preference lists, or upper bounds on the numbers of restricted pairs
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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