290 research outputs found
The Maximum-Weight Stable Matching Problem: Duality and Efficiency
Given a preference system (G,≺) and an integral weight function defined on the edge set of G (not necessarily bipartite), the maximum-weight stable matching problem is to find a stable matching of (G,≺) with maximum total weight. In this paper we study this NP-hard problem using linear programming and polyhedral approaches. We show that the Rothblum system for defining the fractional stable matching polytope of (G,≺) is totally dual integral if and only if this polytope is integral if and only if (G,≺) has a bipartite representation. We also present a combinatorial polynomial-time algorithm for the maximum-weight stable matching problem and its dual on any preference system with a bipartite representation. Our results generalize Király and Pap's theorem on the maximum-weight stable-marriage problem and rely heavily on their work. © 2012 Society for Industrial and Applied Mathematics.published_or_final_versio
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
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
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
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