The stable marriage problem with master preference lists

We study variants of the classical stable marriage problem in which the preferences of the men or the women, or both, are derived from a master preference list. This models real-world matching problems in which participants are ranked according to some objective criteria. The master list(s) may be strictly ordered, or may include ties, and the lists of individuals may involve ties and may include all, or just some, of the members of the opposite sex. In fact, ties are almost inevitable in the master list if the ranking is done on the basis of a scoring scheme with a relatively small range of distinct values. We show that many of the interesting variants of stable marriage that are NP-hard remain so under very severe restrictions involving the presence of master lists, but a number of special cases can be solved in polynomial time. Under this master list model, versions of the stable marriage problem that are already solvable in polynomial time typically yield to faster and/or simpler algorithms, giving rise to simple new structural characterisations of the solutions in these cases

Hard variants of stable marriage

The Stable Marriage Problem and its many variants have been widely studied in the literature (Gusfield and Irving, The Stable Marriage Problem: Structure and Algorithms, MIT Press, Cambridge, MA, 1989; Roth and Sotomayor, Two-sided matching: a study in game-theoretic modeling and analysis, Econometric Society Monographs, vol. 18, Cambridge University Press, Cambridge, 1990; Knuth, Stable Marriage and its Relation to Other Combinatorial Problems, CRM Proceedings and Lecture Notes, vol. 10, American Mathematical Society, Providence, RI, 1997), partly because of the inherent appeal of the problem, partly because of the elegance of the associated structures and algorithms, and partly because of important practical applications, such as the National Resident Matching Program (Roth, J. Political Economy 92(6) (1984) 991) and similar large-scale matching schemes. Here, we present the first comprehensive study of variants of the problem in which the preference lists of the participants are not necessarily complete and not necessarily totally ordered. We show that, under surprisingly restrictive assumptions, a number of these variants are hard, and hard to approximate. The key observation is that, in contrast to the case where preference lists are complete or strictly ordered (or both), a given problem instance may admit stable matchings of different sizes. In this setting, examples of problems that are hard are: finding a stable matching of maximum or minimum size, determining whether a given pair is stable––even if the indifference takes the form of ties on one side only, the ties are at the tails of lists, there is at most one tie per list, and each tie is of length 2; and finding, or approximating, both an `egalitarian' and a `minimum regret' stable matching. However, we give a 2-approximation algorithm for the problems of finding a stable matching of maximum or minimum size. We also discuss the significant implications of our results for practical matching schemes

Robust state estimation using mixed integer programming

This letter describes a robust state estimator based on the solution of a mixed integer program. A tolerance range is associated with each measurement and an estimate is chosen to maximize the number of estimated measurements that remain within tolerance (or equivalently minimize the number of measurements out of tolerance). Some small-scale examples are given which suggest that this approach is robust in the presence of gross errors, is not susceptible to leverage points, and can solve some pathological cases that have previously caused problems for robust estimation algorithms

Pseudo-loadflow formulation as a starting process for the Newton Raphson

This paper introduces new models which approximate the AC loadflow problem, but are able to converge (using the Newton Raphson algorithm) from a wider range of starting points. The solution of the pseudo-loadflow models can provide a robust starting process for the Newton Raphson solution of the conventional loadflow problem. It is also shown that pseudo-loadflow solutions exist in many cases where the AC loadflow equations do not appear to have any solution, and in such cases the pseudo-loadflow solution can provide useful information to assist in locating the cause of infeasibility of the AC loadflow model. Test results are presented for illustrative small network examples and also for larger test networks. The computational requirements of the proposed methods are similar to those of the conventional Newton Raphson loadflow algorithm

Outsourcing the Law: History and the Disciplinary Limits of Constitutional Reasoning

Debates about the use of history in constitutional interpretation find their primary nourishment in the originalism debate. This has generated a vast amount of literature, but also narrowed the terms of the debate. Originalism is a normative commitment wrapped in a questionable methodological confidence. Regardless of the multiple forms originalism takes, originalists are confident that the meaning (in the sense of intention) that animated the framing of the Constitution can be ascertained and, indeed, that they can ascertain it. The debate has largely focused, then, on whether modern-day scholars and jurists can ascertain original historical meaning or, alternatively, whether they have gotten the history right in attempting to do so

Banded Householder representation of linear subspaces

We show how to compactly represent any $n$-dimensional subspace of $R^m$ as a banded product of Householder reflections using $n(m - n)$ floating point numbers. This is optimal since these subspaces form a Grassmannian space $Gr_n(m)$ of dimension $n(m - n)$. The representation is stable and easy to compute: any matrix can be factored into the product of a banded Householder matrix and a square matrix using two to three QR decompositions.Comment: 5 pages, 1 figure, submitted to Linear Algebra and its Application

Paced Auditory Serial Addition Task : an experimental investigation : a thesis presented in partial fulfilment of the requirements for the degree of Master of Arts in Psychology at Massey University

This study investigated the effects of seven independent variables upon a Paced Auditory Serial Addition Task (PASAT; Gronwall & Sampson, 1974). The main effects found were that arithmetic ability and a short-term memory measure were related to performance on this task. Interactive effects were found for measures of anxiety, sex and the strategy used in performing the PASAT. Theories considered included those of Broadbent (1977), Neisser (1976), Kerr (1973), Kahneman (1973) and Broadbent (1971) with emphasis on the latter two. The findings are most easily interpreted in terms of Kahneman's (1973) theory. Broadbent's (1971) model could not account for the effects of environmental and task conditions upon information-processing capacity. Further research is needed to examine the effects of individual abilities and biases in selective attention. Also it is suggested that perception and the allocation of effort policy (Kahneman, 1973) be studied further from Broadbent's (1977) perspective of global and local analysis of information. Clinical implications for the interpretation of the PASAT are discussed. It is suggested that this test could be used more widely as a measure of selective attention. More specifically it is suggested that the administration instructions could be simplified where necessary; and error scores considered together with rate of performance. These measures give an indication of performance effectiveness