Two simple constant ratio approximation algorithms for minimizing the total weighted completion time on a single machine with a fixed non-availability interval

In this note, we consider the scheduling problem of minimizing the sum of the weighted completion times on a single machine with one non-availability interval on the machine under the non-resumable scenario. Together with a recent 2-approximation algorithm designed by Kacem [I. Kacem, Approximation algorithm for the weighted flow-time minimization on a single machine with a fixed non-availability interval, Computers & Industrial Engineering 54 (2008) 401–410], this paper is the first successful attempt to develop a constant ratio approximation algorithm for this problem. We present two approaches to designing such an algorithm. Our best algorithm guarantees a worst-case performance ratio of 2+ε. © 2008 Elsevier B.V. All rights reserved.

We could make a meal out of nothing: contested explanations of inequalities in health

This paper reports on the qualitative findings from a comparative study of public health and lifestyles in South East England and Northern France, regions with similar geographic and economic characteristics. Data from health surveys showed that both countries had an increasing BMI with age, particularly in Northern France. This was despite the finding that the percentage eating fresh fruit and vegetable at least five days a week in Northern France increased with age (from well over 50% to over 90%) compared to around 50% to around 75% in South East England. Qualitative data on health inequalities and how they could be addressed were gathered by focus groups sampling from five tiers using the Townsend Index for comparability (14 in England with 106 participants overall; 13 in France with 143 participants). Both had about two thirds women participants, with a preponderance of middle aged and older people. There was a striking difference in the salience of diet between the two countries; in the French data it was raised only 14 times, whereas in England there were 165 occurrences, and these were often distinguished by their use of narrative. Older respondents contrasted the pressures on families today and the expense of fresh fruit and vegetables with their own childhood or childrearing, when cheap meals could be created using skills which have now been lost. These data therefore provide further evidence that providing food is a moral activity.

Hall conditions for edge-weighted bipartite graphs

A weighted variant of Hall's condition for the existence of matchings is shown to be equivalent to the existence of a matching in a lexicographic product. This is used to introduce characterizations of those bipartite graphs whose edges may be replicated so as to yield semiregular multigraphs or, equivalently, semiregular edge-weightings. Such bipartite graphs will be called semiregularizable. Some infinite families of semiregularizable trees are described and all semiregularizable trees on at most 11 vertices are listed. Matrix analogues of some of the results are mentioned and are shown to imply some of the known characterizations of regularizable graphs.

Selective migration from deprived areas in Northern Ireland and the spatial distribution of inequalities: implications for monitoring health and inequalities in health

Much of the evidence suggesting that inequalities in health have been increasing over the last two decades has come from studies that compared the changes in relative health status of areas over time. Such studies ignore the movement of people between areas. This paper examines the population movement between small areas in Northern Ireland in the year prior to the 1991 census as well as the geographical distribution of migrants to Northern Ireland over the same period. It shows that deprived areas tended to become depopulated and that those who left these areas were the more affluent residents. While immigrants differed a little from the indigenous population, the overall effect of their distribution would be to maintain the geographical socio-economic status quo. The selective movement of people between areas would result in the distribution of health and ill-health becoming more polarized, i.e. produce a picture of widening inequalities between areas even though the distribution between individuals is unchanged. These processes suggest potential significant problems with the area-based approaches to monitoring health and inequalities in health.