How operational research helps kidney patients in the UK

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Keeping partners together: algorithmic results for the hospitals/residents problem with couples

The Hospitals/Residents problem with Couples (HRC) is a generalisation of the classical Hospitals/Residents problem (HR) that is important in practical applications because it models the case where couples submit joint preference lists over pairs of hospitals (h i ,h j ). We consider a natural restriction of HRC in which the members of a couple have individual preference lists over hospitals, and the joint preference list of the couple is consistent with these individual lists in a precise sense. We give an appropriate stability definition and show that, in this context, the problem of deciding whether a stable matching exists is NP-complete, even if each resident’s preference list has length at most 3 and each hospital has capacity at most 2. However, with respect to classical (Gale-Shapley) stability, we give a linear-time algorithm to find a stable matching or report that none exists, regardless of the preference list lengths or the hospital capacities. Finally, for an alternative formulation of our restriction of HRC, which we call the Hospitals/Residents problem with Sizes (HRS), we give a linear-time algorithm that always finds a stable matching for the case that hospital preference lists are of length at most 2, and where hospital capacities can be arbitrary

An 8/5 approximation algorithm for a hard variant of stable marriage

When ties and incomplete preference lists are permitted in the Stable Marriage problem, stable matchings can have different sizes. The problem of finding a maximum cardinality stable matching in this context is NP-hard, even under very severe restrictions on the number, size and position of ties. In this paper, we describe a polynomial-time 8/5-approximation algorithm for a variant in which ties are on one side only and at the end of the preference lists. This variant is motivated by important applications in large scale centralized matching schemes

Estimated Percentage of Females Who Will Become Teen Mothers: Differences Across States

An estimated 18 percent of females nationwide will become teen mothers, according to this Child Trends research brief. The brief also finds that states vary widely in the estimated percentage of females who will have a baby before the age of 20, ranging from 8% in New Hampshire to 30% in Mississippi. Additional findings: --The 2006 estimated percentage of females who will have a teen birth is slightly higher than the 2005 estimate of 17 percent, which reflects a recent increase in the teen birth rate between 2005 and 2006. --For the nation, the estimated percentage of females becoming teen mothers declined from 25 percent in 1991 to 17 percent in 2005, reflecting a drop in teen birth rates during this period. --25 percent or more of females were estimated to become teen mothers in 9 states, concentrated primarily in the South and Southwest. In contrast, in only three states - New Hampshire, Vermont, and Massachusetts - were less than 10 percent of females estimated to become teen mothers. --State-level rankings on the likelihood of becoming a teen mother mirror traditional state rankings based on teen birth rates, but are not identical due to differences in the age-specific birth rates and the population distribution within each state

Paired and altruistic kidney donation in the UK: Algorithms and experimentation

We study the computational problem of identifying optimal sets of kidney exchanges in the UK. We show how to expand an integer programming-based formulation due to Roth et al. [2007] in order to model the criteria that constitute the UK definition of optimality. The software arising from this work has been used by the National Health Service Blood and Transplant to find optimal sets of kidney exchanges for their National Living Donor Kidney Sharing Schemes since July 2008. We report on the characteristics of the solutions that have been obtained in matching runs of the scheme since this time. We then present empirical results arising from experiments on the real datasets that stem from these matching runs, with the aim of establishing the extent to which the particular optimality criteria that are present in the UK influence the structure of the solutions that are ultimately computed. A key observation is that allowing four-way exchanges would be likely to lead to a moderate number of additional transplants

Combined super-/substring and super-/subsequence problems

Super-/substring problems and super-/subsequence problems are well-known problems in stringology that have applications in a variety of areas, such as manufacturing systems design and molecular biology. Here we investigate the complexity of a new type of such problem that forms a combination of a super-/substring and a super-/subsequence problem. Moreover we introduce different types of minimal superstring and maximal substring problems. In particular, we consider the following problems: given a set L of strings and a string S, (i) find a minimal superstring (or maximal substring) of L that is also a supersequence (or a subsequence) of S, (ii) find a minimal supersequence (or maximal subsequence) of L that is also a superstring (or a substring) of S. In addition some non-super-/non-substring and non-super-/non-subsequence variants are studied. We obtain several NP-hardness or even MAX SNP-hardness results and also identify types of "weak minimal" superstrings and "weak maximal" substrings for which (i) is polynomial-time solvable

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

Condom Use and Consistency Among Teen Males

This Child Trends study finds that any type of formal sex education is linked with higher levels of condom use at teen males' first sexual experience. However, one in five teen males (ages 15-19) did not receive formal sex education about either abstinence or contraception before having sex for the first time. The study, published in the October 2008 issue of the Journal of Adolescent Health and summarized in this fact sheet, examines how multiple dimensions of teen males' lives are associated with condom use and consistency. Among the findings: --Having an older partner or a casual partner is linked to less condom use. Nearly one-fourth of teen males had an older recent partner and more than one-third were in a casual relationship with their first sexual partner. --Older teen males and those in longer relationships are less likely to use condoms. This was true even after controlling for whether their partner used a contraceptive method. --Positive attitudes about using condoms are linked to actual use. Teen males who disagree with the ideas that condoms reduce physical pleasure and that it would be embarrassing to discuss condom use with a new partner have higher levels of condom use and consistency. --Seven in ten teen males reported using a condom at their first and at their most recent sexual experience, but fewer reported using condoms consistently. Just one-half of sexually active teen males reported using a condom consistently with their most recent sexual partner

Popular matchings in the weighted capacitated house allocation problem

We consider the problem of finding a popular matching in the <i>Weighted Capacitated House Allocation</i> problem (WCHA). An instance of WCHA involves a set of agents and a set of houses. Each agent has a positive weight indicating his priority, and a preference list in which a subset of houses are ranked in strict order. Each house has a capacity that indicates the maximum number of agents who could be matched to it. A matching M of agents to houses is popular if there is no other matching M′ such that the total weight of the agents who prefer their allocation in M′ to that in M exceeds the total weight of the agents who prefer their allocation in M to that in M′. Here, we give an [FORMULA] algorithm to determine if an instance of WCHA admits a popular matching, and if so, to find a largest such matching, where C is the total capacity of the houses, n1 is the number of agents, and m is the total length of the agents' preference lists