191 research outputs found
Integer programming methods for special college admissions problems
We develop Integer Programming (IP) solutions for some special college
admission problems arising from the Hungarian higher education admission
scheme. We focus on four special features, namely the solution concept of
stable score-limits, the presence of lower and common quotas, and paired
applications. We note that each of the latter three special feature makes the
college admissions problem NP-hard to solve. Currently, a heuristic based on
the Gale-Shapley algorithm is being used in the application. The IP methods
that we propose are not only interesting theoretically, but may also serve as
an alternative solution concept for this practical application, and also for
other ones
The weighted stable matching problem
We study the stable matching problem in non-bipartite graphs with incomplete
but strict preference lists, where the edges have weights and the goal is to
compute a stable matching of minimum or maximum weight. This problem is known
to be NP-hard in general. Our contribution is two fold: a polyhedral
characterization and an approximation algorithm. Previously Chen et al. have
shown that the stable matching polytope is integral if and only if the subgraph
obtained after running phase one of Irving's algorithm is bipartite. We improve
upon this result by showing that there are instances where this subgraph might
not be bipartite but one can further eliminate some edges and arrive at a
bipartite subgraph. Our elimination procedure ensures that the set of stable
matchings remains the same, and thus the stable matching polytope of the final
subgraph contains the incidence vectors of all stable matchings of our original
graph. This allows us to characterize a larger class of instances for which the
weighted stable matching problem is polynomial-time solvable. We also show that
our edge elimination procedure is best possible, meaning that if the subgraph
we arrive at is not bipartite, then there is no bipartite subgraph that has the
same set of stable matchings as the original graph. We complement these results
with a -approximation algorithm for the minimum weight stable matching
problem for instances where each agent has at most two possible partners in any
stable matching. This is the first approximation result for any class of
instances with general weights.Comment: This is an extended version of a paper to appear at the The Fourth
International Workshop on Matching Under Preferences (MATCH-UP 2017
Classified Stable Matching
We introduce the {\sc classified stable matching} problem, a problem
motivated by academic hiring. Suppose that a number of institutes are hiring
faculty members from a pool of applicants. Both institutes and applicants have
preferences over the other side. An institute classifies the applicants based
on their research areas (or any other criterion), and, for each class, it sets
a lower bound and an upper bound on the number of applicants it would hire in
that class. The objective is to find a stable matching from which no group of
participants has reason to deviate. Moreover, the matching should respect the
upper/lower bounds of the classes.
In the first part of the paper, we study classified stable matching problems
whose classifications belong to a fixed set of ``order types.'' We show that if
the set consists entirely of downward forests, there is a polynomial-time
algorithm; otherwise, it is NP-complete to decide the existence of a stable
matching.
In the second part, we investigate the problem using a polyhedral approach.
Suppose that all classifications are laminar families and there is no lower
bound. We propose a set of linear inequalities to describe stable matching
polytope and prove that it is integral. This integrality allows us to find
various optimal stable matchings using Ellipsoid algorithm. A further
ramification of our result is the description of the stable matching polytope
for the many-to-many (unclassified) stable matching problem. This answers an
open question posed by Sethuraman, Teo and Qian
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
An Algorithm for the Maximum Weight Strongly Stable Matching Problem
An instance of the maximum weight strongly stable matching problem with incomplete lists and ties is an undirected bipartite graph G = (A cup B, E), with an adjacency list being a linearly ordered list of ties, which are vertices equally good for a given vertex. We are also given a weight function w on the set E. An edge (x, y) in E setminus M is a blocking edge for M if by getting matched to each other neither of the vertices x and y would become worse off and at least one of them would become better off. A matching is strongly stable if there is no blocking edge with respect to it. The goal is to compute a strongly stable matching of maximum weight with respect to w.
We give a polyhedral characterisation of the problem and prove that the strongly stable matching polytope is integral. This result implies that the maximum weight strongly stable matching problem can be solved in polynomial time. Thereby answering an open question by Gusfield and Irving [Dan Gusfield and Robert W. Irving, 1989]. The main result of this paper is an efficient O(nm log{(Wn)}) time algorithm for computing a maximum weight strongly stable matching, where we denote n = |V|, m = |E| and W is a maximum weight of an edge in G. For small edge weights we show that the problem can be solved in O(nm) time. Note that the fastest known algorithm for the unweighted version of the problem has O(nm) runtime [Telikepalli Kavitha et al., 2007]. Our algorithm is based on the rotation structure which was constructed for strongly stable matchings in [Adam Kunysz et al., 2016]
Stable matchings and linear programming
AbstractThis paper continues the work of Abeledo and Rothblum, who study nonbipartite stable matching problems from a polyhedral perspective. We establish here additional properties of fractional stable matchings and use linear programming to obtain an alternative polynomial algorithm for solving stable matching problems
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