470 research outputs found

    Approximation algorithms for hard variants of the stable marriage and hospitals/residents problems

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    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

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    Given a set of nn men represented by nn points lying on a line, and nn women represented by nn 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 MM is a pair (m,w)(m,w) of a man and a woman such that they are not matched with each other but prefer each other to their own partners in MM, and the segment (m,w)(m,w) does not cross any edge in MM. 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 O(n2)O(n^2) algorithm to find one in a given instance.Comment: This paper has appeared at IWOCA 201

    Manipulation Strategies for the Rank Maximal Matching Problem

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    We consider manipulation strategies for the rank-maximal matching problem. In the rank-maximal matching problem we are given a bipartite graph G=(AP,E)G = (A \cup P, E) such that AA denotes a set of applicants and PP a set of posts. Each applicant aAa \in A has a preference list over the set of his neighbours in GG, possibly involving ties. Preference lists are represented by ranks on the edges - an edge (a,p)(a,p) has rank ii, denoted as rank(a,p)=irank(a,p)=i, if post pp belongs to one of aa's ii-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 O(min(cn,n)m)O(\min(c \sqrt{n},n) m) time, where nn denotes the number of applicants, mm the number of edges and cc 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 GG, we assume that the central authority chooses any one of them randomly. Let a1a_1 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 a1a_1 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, a1a_1 wants to minimize the maximal rank of a post he can become matched to

    The Stable Roommates problem with short lists

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    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

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    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

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    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

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    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

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    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>

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    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

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    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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