470 research outputs found
Approximation algorithms for hard variants of the stable marriage and hospitals/residents problems
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 paper, we describe polynomial-time 5/3-approximation algorithms for variants of these problems in which ties are on one side only and at the end of the preference lists. The particular variant is motivated by important applications in large scale centralised matching schemes
Stable Noncrossing Matchings
Given a set of men represented by points lying on a line, and
women represented by points lying on another parallel line, with each
person having a list that ranks some people of opposite gender as his/her
acceptable partners in strict order of preference. In this problem, we want to
match people of opposite genders to satisfy people's preferences as well as
making the edges not crossing one another geometrically. A noncrossing blocking
pair w.r.t. a matching is a pair of a man and a woman such that
they are not matched with each other but prefer each other to their own
partners in , and the segment does not cross any edge in . A
weakly stable noncrossing matching (WSNM) is a noncrossing matching that does
not admit any noncrossing blocking pair. In this paper, we prove the existence
of a WSNM in any instance by developing an algorithm to find one in a
given instance.Comment: This paper has appeared at IWOCA 201
Manipulation Strategies for the Rank Maximal Matching Problem
We consider manipulation strategies for the rank-maximal matching problem. In
the rank-maximal matching problem we are given a bipartite graph such that denotes a set of applicants and a set of posts. Each
applicant has a preference list over the set of his neighbours in
, possibly involving ties. Preference lists are represented by ranks on the
edges - an edge has rank , denoted as , if post
belongs to one of 's -th choices. A rank-maximal matching is one in which
the maximum number of applicants is matched to their rank one posts and subject
to this condition, the maximum number of applicants is matched to their rank
two posts, and so on. A rank-maximal matching can be computed in time, where denotes the number of applicants, the
number of edges and the maximum rank of an edge in an optimal solution.
A central authority matches applicants to posts. It does so using one of the
rank-maximal matchings. Since there may be more than one rank- maximal matching
of , we assume that the central authority chooses any one of them randomly.
Let be a manipulative applicant, who knows the preference lists of all
the other applicants and wants to falsify his preference list so that he has a
chance of getting better posts than if he were truthful. In the first problem
addressed in this paper the manipulative applicant wants to ensure that
he is never matched to any post worse than the most preferred among those of
rank greater than one and obtainable when he is truthful. In the second problem
the manipulator wants to construct such a preference list that the worst post
he can become matched to by the central authority is best possible or in other
words, wants to minimize the maximal rank of a post he can become matched
to
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,
EGAL d-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+3)/7-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.Comment: short version appeared at SAGT 201
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
New and simple algorithms for stable flow problems
Stable flows generalize the well-known concept of stable matchings to markets
in which transactions may involve several agents, forwarding flow from one to
another. An instance of the problem consists of a capacitated directed network,
in which vertices express their preferences over their incident edges. A
network flow is stable if there is no group of vertices that all could benefit
from rerouting the flow along a walk.
Fleiner established that a stable flow always exists by reducing it to the
stable allocation problem. We present an augmenting-path algorithm for
computing a stable flow, the first algorithm that achieves polynomial running
time for this problem without using stable allocation as a black-box
subroutine. We further consider the problem of finding a stable flow such that
the flow value on every edge is within a given interval. For this problem, we
present an elegant graph transformation and based on this, we devise a simple
and fast algorithm, which also can be used to find a solution to the stable
marriage problem with forced and forbidden edges.
Finally, we study the stable multicommodity flow model introduced by
Kir\'{a}ly and Pap. The original model is highly involved and allows for
commodity-dependent preference lists at the vertices and commodity-specific
edge capacities. We present several graph-based reductions that show
equivalence to a significantly simpler model. We further show that it is
NP-complete to decide whether an integral solution exists
Group Strategyproof Pareto-Stable Marriage with Indifferences via the Generalized Assignment Game
We study the variant of the stable marriage problem in which the preferences
of the agents are allowed to include indifferences. We present a mechanism for
producing Pareto-stable matchings in stable marriage markets with indifferences
that is group strategyproof for one side of the market. Our key technique
involves modeling the stable marriage market as a generalized assignment game.
We also show that our mechanism can be implemented efficiently. These results
can be extended to the college admissions problem with indifferences
Rotation-based formulation for stable matching
We introduce new CP models for the many-to-many stable matching problem. We use the notion of rotation to give a novel encoding that is linear in the input size of the problem. We give extra filtering rules to maintain arc consistency in quadratic time. Our experimental study on hard instances of sex-equal and balanced stable matching shows the efficiency of one of our propositions as compared with the state-of-the-art constraint programming approach
Defects, Dopants and Sodium Mobility in Na<sub>2</sub>MnSiO<sub>4</sub>
Sodium manganese orthosilicate, Na2MnSiO4, is a promising positive electrode material in rechargeable sodium ion batteries. Atomistic scale simulations are used to study the defects, doping behaviour and sodium migration paths in Na2MnSiO4. The most favourable intrinsic defect type is the cation anti-site (0.44 eV/defect), in which, Na and Mn exchange their positions. The second most favourable defect energy process is found to be the Na Frenkel (1.60 eV/defect) indicating that Na diffusion is assisted by the formation of Na vacancies via the vacancy mechanism. Long range sodium paths via vacancy mechanism were constructed and it is confirmed that the lowest activation energy (0.81 eV) migration path is three dimensional with zig-zag pattern. Subvalent doping by Al on the Si site is energetically favourable suggesting that this defect engineering stratergy to increase the Na content in Na2MnSiO4 warrants experimental verification
Hand Grip Strength: age and gender stratified normative data in a population-based study
Extent: 5p.Background: The North West Adelaide Health Study is a representative longitudinal cohort study of people originally aged 18 years and over. The aim of this study was to describe normative data for hand grip strength in a community-based Australian population. Secondary aims were to investigate the relationship between body mass index (BMI) and hand grip strength, and to compare Australian data with international hand grip strength norms. Methods: The sample was randomly selected and recruited by telephone interview. Overall, 3 206 (81% of those recruited) participants returned to the clinic during the second stage (2004-2006) which specifically focused on the collection of information relating to musculoskeletal conditions. Results: Following the exclusion of 435 participants who had hand pain and/or arthritis, 1366 men and 1312 women participants provided hand grip strength measurement. The study population was relatively young, with 41.5% under 40 years; and their mean BMI was 28.1 kg/m2 (SD 5.5). Higher hand grip strength was weakly related to higher BMI in adults under the age of 30 and over the age of 70, but inversely related to higher BMI between these ages. Australian norms from this sample had amongst the lowest of the hand grip strength of the internationally published norms, except those from underweight populations. Conclusions: This population demonstrated higher BMI and lower grip strength in younger participants than much of the international published, population data. A complete exploration of the relationship between BMI and hand grip strength was not fully explored as there were very few participants with BMI in the underweight range. The age and gender grip strength values are lower in younger adults than those reported in international literature.Nicola M Massy-Westropp, Tiffany K Gill, Anne W Taylor, Richard W Bohannon and Catherine L Hil
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