14,364 research outputs found
Computing Optimal Morse Matchings
Morse matchings capture the essential structural information of discrete
Morse functions. We show that computing optimal Morse matchings is NP-hard and
give an integer programming formulation for the problem. Then we present
polyhedral results for the corresponding polytope and report on computational
results
Pareto optimality in house allocation problems
We study Pareto optimal matchings in the context of house allocation problems. We present an O(\sqrt{n}m) algorithm, based on Gales Top Trading Cycles Method, for finding a maximum cardinality Pareto optimal matching, where n is the number of agents and m is the total length of the preference lists. By contrast, we show that the problem of finding a minimum cardinality Pareto optimal matching is NP-hard, though approximable within a factor of 2. We then show that there exist Pareto optimal matchings of all sizes between a minimum and maximum cardinality Pareto optimal matching. Finally, we introduce the concept of a signature, which allows us to give a characterization, checkable in linear time, of instances that admit a unique Pareto optimal matching
Large matchings in uniform hypergraphs and the conjectures of Erdos and Samuels
In this paper we study conditions which guarantee the existence of perfect
matchings and perfect fractional matchings in uniform hypergraphs. We reduce
this problem to an old conjecture by Erd\H{o}s on estimating the maximum number
of edges in a hypergraph when the (fractional) matching number is given, which
we are able to solve in some special cases using probabilistic techniques.
Based on these results, we obtain some general theorems on the minimum
-degree ensuring the existence of perfect (fractional) matchings. In
particular, we asymptotically determine the minimum vertex degree which
guarantees a perfect matching in 4-uniform and 5-uniform hypergraphs. We also
discuss an application to a problem of finding an optimal data allocation in a
distributed storage system
On the dimension of the core of the assignment game
The set of optimal matchings in the assignment matrix allows to define a reflexive and symmetric binary relation on each side of the market, the equal-partner binary relation. The number of equivalence classes of the transitive closure of the equal-partner binary relation determines the dimension of the core of the assignment game. This result provides an easy procedure to determine the dimension of the core directly from the entries of the assignment matrix and shows that the dimension of the core is not as much determined by the number of optimal matchings as by their relative position in the assignment matrix.core, assignment game, core dimension
Approximating Semi-Matchings in Streaming and in Two-Party Communication
We study the communication complexity and streaming complexity of
approximating unweighted semi-matchings. A semi-matching in a bipartite graph G
= (A, B, E), with n = |A|, is a subset of edges S that matches all A vertices
to B vertices with the goal usually being to do this as fairly as possible.
While the term 'semi-matching' was coined in 2003 by Harvey et al. [WADS 2003],
the problem had already previously been studied in the scheduling literature
under different names.
We present a deterministic one-pass streaming algorithm that for any 0 <=
\epsilon <= 1 uses space O(n^{1+\epsilon}) and computes an
O(n^{(1-\epsilon)/2})-approximation to the semi-matching problem. Furthermore,
with O(log n) passes it is possible to compute an O(log n)-approximation with
space O(n).
In the one-way two-party communication setting, we show that for every
\epsilon > 0, deterministic communication protocols for computing an
O(n^{1/((1+\epsilon)c + 1)})-approximation require a message of size more than
cn bits. We present two deterministic protocols communicating n and 2n edges
that compute an O(sqrt(n)) and an O(n^{1/3})-approximation respectively.
Finally, we improve on results of Harvey et al. [Journal of Algorithms 2006]
and prove new links between semi-matchings and matchings. While it was known
that an optimal semi-matching contains a maximum matching, we show that there
is a hierarchical decomposition of an optimal semi-matching into maximum
matchings. A similar result holds for semi-matchings that do not admit
length-two degree-minimizing paths.Comment: This is the long version including all proves of the ICALP 2013 pape
Matching with Trade-offs: Revealed Preferences over Competiting Characteristics
We investigate in this paper the theory and econometrics of optimal matchings with competing criteria. The surplus from a marriage match, for instance, may depend both on the incomes and on the educations of the partners, as well as on characteristics that the analyst does not observe. The social optimum must therefore trade off matching on incomes and matching on educations. Given a exible specification of the surplus function, we characterize under mild assumptions the properties of the set of feasible matchings and of the socially optimal matching. Then we show how data on the covariation of the types of the partners in observed matches can be used to estimate the parameters that define social preferences over matches. We provide both nonparametric and parametric procedures that are very easy to use in applications.matching, marriage, assignment.
Algorithms for New Types of Fair Stable Matchings
We study the problem of finding "fair" stable matchings in the Stable Marriage problem with Incomplete lists (SMI). For an instance I of SMI there may be many stable matchings, providing significantly different outcomes for the sets of men and women. We introduce two new notions of fairness in SMI. Firstly, a regret-equal stable matching minimises the difference in ranks of a worst-off man and a worst-off woman, among all stable matchings. Secondly, a min-regret sum stable matching minimises the sum of ranks of a worst-off man and a worst-off woman, among all stable matchings. We present two new efficient algorithms to find stable matchings of these types. Firstly, the Regret-Equal Degree Iteration Algorithm finds a regret-equal stable matching in O(d0nm) time, where d0 is the absolute difference in ranks between a worst-off man and a worst-off woman in the man-optimal stable matching, n is the number of men or women, and m is the total length of all preference lists. Secondly, the Min-Regret Sum Algorithm finds a min-regret sum stable matching in O(dsm) time, where ds is the difference in the ranks between a worst-off man in each of the woman-optimal and man-optimal stable matchings. Experiments to compare several types of fair optimal stable matchings were conducted and show that the Regret-Equal Degree Iteration Algorithm produces matchings that are competitive with respect to other fairness objectives. On the other hand, existing types of "fair" stable matchings did not provide as close an approximation to regret-equal stable matchings
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