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

Collision of High Frequency Plane Gravitational and Electromagnetic Waves

We study the head-on collision of linearly polarized, high frequency plane gravitational waves and their electromagnetic counterparts in the Einstein-Maxwell theory. The post-collision space-times are obtained by solving the vacuum Einstein-Maxwell field equations in the geometrical optics approximation. The head-on collisions of all possible pairs of these systems of waves is described and the results are then generalised to non-linearly polarized waves which exhibit the maximum two degrees of freedom of polarization.Comment: Latex file, 17 pages, accepted for publication in International Journal of Modern Physics

Income-Related Subsidies for Universal Health Insurance Premia: Exploring Alternatives Using the SWITCH Model. ESRI WP516. November 2015

The Programme for Government indicated that under a Universal Health Insurance system, the State would “pay insurance premia for people on low incomes and subsidise premia for people on middle incomes”. This paper examines issues in the design of such a subsidy scheme, in the context of overall premium costs as estimated by Wren et al. (2015) and the KPMG (2015) study for the Health Insurance Authority. Subsidy design could involve a step-level system, similar to the medical card and GP visit card in the current system; or a smooth, tapered withdrawal of the subsidy, similar to what obtains for many cash benefits in the welfare system. The trade-offs between the income limit up to which a full subsidy would be payable, the rate of withdrawal of subsidy with respect to extra income and overall subsidy cost are explored

On The Complexity and Completeness of Static Constraints for Breaking Row and Column Symmetry

We consider a common type of symmetry where we have a matrix of decision variables with interchangeable rows and columns. A simple and efficient method to deal with such row and column symmetry is to post symmetry breaking constraints like DOUBLELEX and SNAKELEX. We provide a number of positive and negative results on posting such symmetry breaking constraints. On the positive side, we prove that we can compute in polynomial time a unique representative of an equivalence class in a matrix model with row and column symmetry if the number of rows (or of columns) is bounded and in a number of other special cases. On the negative side, we show that whilst DOUBLELEX and SNAKELEX are often effective in practice, they can leave a large number of symmetric solutions in the worst case. In addition, we prove that propagating DOUBLELEX completely is NP-hard. Finally we consider how to break row, column and value symmetry, correcting a result in the literature about the safeness of combining different symmetry breaking constraints. We end with the first experimental study on how much symmetry is left by DOUBLELEX and SNAKELEX on some benchmark problems.Comment: To appear in the Proceedings of the 16th International Conference on Principles and Practice of Constraint Programming (CP 2010

The Phase Diagram of 1-in-3 Satisfiability Problem

We study the typical case properties of the 1-in-3 satisfiability problem, the boolean satisfaction problem where a clause is satisfied by exactly one literal, in an enlarged random ensemble parametrized by average connectivity and probability of negation of a variable in a clause. Random 1-in-3 Satisfiability and Exact 3-Cover are special cases of this ensemble. We interpolate between these cases from a region where satisfiability can be typically decided for all connectivities in polynomial time to a region where deciding satisfiability is hard, in some interval of connectivities. We derive several rigorous results in the first region, and develop the one-step--replica-symmetry-breaking cavity analysis in the second one. We discuss the prediction for the transition between the almost surely satisfiable and the almost surely unsatisfiable phase, and other structural properties of the phase diagram, in light of cavity method results.Comment: 30 pages, 12 figure

TACKLING LOW INCOME AND DEPRIVATION: DEVELOPING EFFECTIVE POLICIES. RESEARCH SERIES NUMBER 1 JUNE 2008

This study examines options for the future development of the Irish welfare state, with a view to tackling low income and deprivation more effectively. It draws on existing national and cross-country evidence as well as providing new analyses to inform policy debate in this area. The impact of alternative policy choices is explored using SWITCH, the ESRI tax-benefit model

Stochastic theory of large-scale enzyme-reaction networks: Finite copy number corrections to rate equation models

Chemical reactions inside cells occur in compartment volumes in the range of atto- to femtolitres. Physiological concentrations realized in such small volumes imply low copy numbers of interacting molecules with the consequence of considerable fluctuations in the concentrations. In contrast, rate equation models are based on the implicit assumption of infinitely large numbers of interacting molecules, or equivalently, that reactions occur in infinite volumes at constant macroscopic concentrations. In this article we compute the finite-volume corrections (or equivalently the finite copy number corrections) to the solutions of the rate equations for chemical reaction networks composed of arbitrarily large numbers of enzyme-catalyzed reactions which are confined inside a small sub-cellular compartment. This is achieved by applying a mesoscopic version of the quasi-steady state assumption to the exact Fokker-Planck equation associated with the Poisson Representation of the chemical master equation. The procedure yields impressively simple and compact expressions for the finite-volume corrections. We prove that the predictions of the rate equations will always underestimate the actual steady-state substrate concentrations for an enzyme-reaction network confined in a small volume. In particular we show that the finite-volume corrections increase with decreasing sub-cellular volume, decreasing Michaelis-Menten constants and increasing enzyme saturation. The magnitude of the corrections depends sensitively on the topology of the network. The predictions of the theory are shown to be in excellent agreement with stochastic simulations for two types of networks typically associated with protein methylation and metabolism.Comment: 13 pages, 4 figures; published in The Journal of Chemical Physic

Modelling Eligibility for Medical Cards and GP Visit Cards: Methods and Baseline Results. ESRI WP515. November 2015

The Irish healthcare system includes a complex mix of entitlements – some are universal, others age-related, and some are income-related. In this report, we concentrate on the major income-related entitlements in the current system i.e., the Medical Card and the GP Visit Card. Most medical cards are provided on an income-tested basis, and provide free access to in-patient and out-patient care in public hospitals, to GP care, and to prescription drugs. We examine how the income test for such schemes can be modelled using the detailed income and demographic information in the Survey on Income and Living Conditions. The approach taken applies the rules for income-related cards to each family in this nationally representative sample, using the information they provide on incomes and family composition. This is essential groundwork for later studies which will examine how the pattern of entitlements might change under different rules, such as those introducing age-related entitlements to GP visit cards, or changes in income limits

BUDGET PERSPECTIVES 2016, PAPER 1. Exploring Tax and Welfare Options. June 2015

Budgetary policies on income-related taxes and welfare must find a balance between providing income support to those in need and maintaining a financial incentive to work which supports high employment. This paper focuses principally on the “cash” or “first round” impact of tax and welfare policy changes across the income distribution. Incentive issues are considered in Section 5 of this paper, and in a companion paper to this conference (Savage et al., 2015)

BUDGET PERSPECTIVES 2016, PAPER 2. Making Work Pay More: Recent Initiatives. June 2015

This paper examines the financial incentives to work implicit in the Irish tax and benefit system, focusing in particular on incentives facing those who are unemployed and in receipt of Jobseeker’s Benefit or Jobseeker’s Assistance. The results, based on an analysis of current incomes, benefits and taxes, suggest that more than eight out of ten of these unemployed jobseekers would see their income increase by at least 40 per cent upon taking up employment. Fewer than 3 per cent of these individuals would, in the short-term, be financially better off not in work. The risk of facing weak financial incentives to work is higher for unemployed persons with a spouse and children, as the income support goal of the welfare system means that they tend to have higher welfare payments. However, even among that group, fewer than 1 in 15 would be financially better off not working. Our analysis shows that a recent policy initiative, the Back to Work Family Dividend, announced in Budget 2015, clearly improves the immediate financial incentives to work for this group