Modelling and Solving the Stable Marriage Problem Using Constraint Programming

We study the Stable Marriage problem (SM), which is a combinatorial problem that arises in many practical applications. We present two new models of an instance I of SM with n men and n women as an instance J of a Constraint Satisfaction Problem. We prove that establishing arc consistency in J yields the same structure as given by the established Extended Gale/Shapley algorithm for SM as applied to I. Consequently, a solution (stable matching) of I can be derived without search. Furthermore we show that, in both encodings, all stable matchings in I may be enumerated in a failure-free manner. Our first encoding is of O(n^3) complexity and is very natural, whilst our second model, of O(n^2) complexity (which is optimal), is a development of the Boolean encoding in [6], establishing a greater level of structure

Local search for stable marriage problems

The stable marriage (SM) problem has a wide variety of practical applications, ranging from matching resident doctors to hospitals, to matching students to schools, or more generally to any two-sided market. In the classical formulation, n men and n women express their preferences (via a strict total order) over the members of the other sex. Solving a SM problem means finding a stable marriage where stability is an envy-free notion: no man and woman who are not married to each other would both prefer each other to their partners or to being single. We consider both the classical stable marriage problem and one of its useful variations (denoted SMTI) where the men and women express their preferences in the form of an incomplete preference list with ties over a subset of the members of the other sex. Matchings are permitted only with people who appear in these lists, an we try to find a stable matching that marries as many people as possible. Whilst the SM problem is polynomial to solve, the SMTI problem is NP-hard. We propose to tackle both problems via a local search approach, which exploits properties of the problems to reduce the size of the neighborhood and to make local moves efficiently. We evaluate empirically our algorithm for SM problems by measuring its runtime behaviour and its ability to sample the lattice of all possible stable marriages. We evaluate our algorithm for SMTI problems in terms of both its runtime behaviour and its ability to find a maximum cardinality stable marriage.For SM problems, the number of steps of our algorithm grows only as O(nlog(n)), and that it samples very well the set of all stable marriages. It is thus a fair and efficient approach to generate stable marriages.Furthermore, our approach for SMTI problems is able to solve large problems, quickly returning stable matchings of large and often optimal size despite the NP-hardness of this problem.Comment: 12 pages, Proc. COMSOC 2010 (Third International Workshop on Computational Social Choice

Vascular risk factors in the Swiss population

Background and Purpose : Identification of the population at risk of stroke remains the best approach to assess the burden of cardiovascular morbidity and mortality. Methods : The prevalence of hypertension (HT), hypercholesterolemia (HCh), diabetes mellitus (DM), overweight (OW), obesity (OB), tobacco use (SM), and their combinations was examined in 4458 Swiss persons (1741 men and 2717 women, mean age 57.8 ± 15 years), who volunteered for the present survey. Results : OW was the most prevalent risk factor (50 %), followed by HT (47%), HCh (33%), SM (13 %) and DM (1.6 %). The proportion of persons without risk factors (RF) was 19.9%, with 1 RF 41.5%, 2 RF 33.8%, 3 RF 4%, and 4 RF 0.9%. OW was more prevalent in men than in women (53% vs. 41%, P=0.02). More men than women aged 41-50 years and 51-60 years had HT (49 % vs. 36%, P=0.01, and 52 % vs. 42%, P=0.02). The prevalence of HCh and DM did not show any sex-related differences. HT, OW and HCh were not only the most common single risk factors, but were also most likely to aggregate with each other. Conclusions : The majority of Swiss people have one or two vascular risk factors. OW and HT are by far most common and are likely to aggregate with each other. A small modification of these two factors would reduce the incidence of stroke and myocardial infarction significantl

Validity of segmental bioelectrical impedance method for body composition assessment in Japanese athletes

The purpose of this study was to investigate the validation of body composition assessment method using segmental bioelectrical impedance (S-BIA) method compared with dual energy X-ray absorptiometry (DEXA) method as gold standard method in college athletes. This study used S-BIA (Physion MD, Physion Co. Ltd., Japan) and DEXA (DPX-L, Lunar Co. Ltd., USA). The subjects were 16 athletes (men: 11, women: 5) belonging to the university track and field club. We measured the body composition in a supine state using S-BIA method. The measurement by S-BIA method was fat mass (FM), percent of body fat (%FAT), fat free mass (FFM) in whole body, and skeletal muscle mass (SM) in arms, legs and trunk. In addition, using the DEXA method, we evaluated FM, %Fat, FFM, and SM in the right and left of limbs and trunk. The LTM by the DEXA method was as an index of the SM. The result was as follows. The FFM, FM and %FAT in whole body by S-BIA method were high correlated with the values measured by DXA (p＜0.001–0.05)．In FFM, FM and %FAT there is no systematic error by examination of the validity by the Bland-Altman method, and there are all data in 95% limits of agreement. The SM of each segment by S-BIA method was significantly correlated with the value by DEXA method, and possibility to be able to perform the evaluation of the SM in athletes. Therefore it was suggested that S-BIA method was effective for an evaluation of the whole and segmental body composition in athletes

Reliability of Multisensor Armband in Estimating Energy Expenditure According to Degree of Obesity

Resting energy expenditure (REE) represents the amount of calories required by the body to maintain vital bodyfunctions. One of the most commonly used methods for estimating REE is indirect calorimetry. Recent studies ondifferent populations have validated a highly innovative instrument, the SenseWear® Armband (SWA), which evaluatestotal energy expenditure and, when used in resting conditions, could also evaluate REE. The purpose of this study wasto determine the agreement of the SWA in assessing REE in obese subjects and, see how this agreement varies withdifferent obesity degree.89 obese subjects (59 women and 30 men), with an age range from 35-65 years and body mass index (BMI)34.5 4.5 kg/m2 were studied. REE was measured by IC Sensor Medics Vmax (SM-29N) and by SWA. Fat mass(FM) and fat-free mass (FFM) was determined by anthropometry and bio-impedance measurements. No statisticaldifference was found between REE measured by SWA (1693±276) and REE measured by SM-29N (1627±293). Thetwo methods showed similar assessments (r=0.8, p 35 kg/m2), the agreement decreases (r =0.6 p 35

The exchange-stable marriage problem

In this paper we consider instances of stable matching problems, namely the classical stable marriage (SM) and stable roommates (SR) problems and their variants. In such instances we consider a stability criterion that has recently been proposed, that of <i>exchange-stability</i>. In particular, we prove that ESM — the problem of deciding, given an SM instance, whether an exchange-stable matching exists — is NP-complete. This result is in marked contrast with Gale and Shapley's classical linear-time algorithm for finding a stable matching in an instance of SM. We also extend the result for ESM to the SR case. Finally, we study some variants of ESM under weaker forms of exchange-stability, presenting both polynomial-time solvability and NP-completeness results for the corresponding existence questions

Single-Father Families: A Review of the Literature

The number of children residing in single-father families in the United States quadrupled as a proportion of children\u27s living arrangements during the past few decades of the 20th century. Research on single fathers also increased and changed in nature. This article is a review of the research on single fathers and their families from the 1970s until recently, focusing on modifications in methodology and theoretical underpinnings. In general, research on single-father families evolved from qualitative studies focused on the well-being of single fathers to quantitative studies focusing on child outcomes and within-group variation among single fathers. Research also moved from descriptive studies to those testing gender and microstructural theories. This article also summarizes the main findings on single fathers and concludes with directions for future research