A new solution for the roommate problem: The Q-stable matchings

The aim of this paper is to propose a new solution for the roommate problem with strict preferences. We introduce the solution of maximum irreversibility and consider almost stable matchings (Abraham et al. [2])and maximum stable matchings (Ta [30] [32]). We find that almost stable matchings are incompatible with the other two solutions. Hence, to solve the roommate problem we propose matchings that lie at the intersection of the maximum irreversible matchings and maximum stable matchings, which are called Q-stable matchings. These matchings are core consistent and we offer an effi cient algorithm for computing one of them. The outcome of the algorithm belongs to an absorbing set.This research is supported by the Spanish Ministry of Science and Innovation (ECO2010- 17049 and ECO2012-31346), co-funded by ERDF, by Basque Government IT-568-13 and by the Government of Andalusia Project for Excellence in Research (P07.SEJ.02547). P eter Bir o also acknowledges the support from the Hungarian Academy of Sciences under its Momentum Programme (LD-004/2010), and the Hungarian Scientific Research Fund,OTKA, grant no.K108673

A new solution concept for the roommate problem

Abstract The aim of this paper is to propose a new solution concept for the roommate problem with strict preferences. We introduce maximum irreversible matchings and consider almost stable matchings (Abraham et al., 2006) and maximum stable matchings (Tan 1990, 1991b). These solution concepts are all core consistent. We find that almost stable matchings are incompatible with the other two concepts. Hence, to solve the roommate problem we propose matchings that lie at the intersection of the maximum irreversible matchings and maximum stable matchings, which we call Q -stable matchings. We construct an efficient algorithm for computing one element of this set for any roommate problem. We also show that the outcome of our algorithm always belongs to an absorbing set (Inarra et al., 2013)

A new solution for the roommate problem

The aim of this paper is to propose a new solution for the roommate problem with strict references. We introduce the solution of maximum ir reversibility and consider almost stable matchings (Abraham et al. [2]) and maximum stable m atchings (Tan [30] [32]). We find that almost stable matchings are incompatible with the o ther two solutions. Hence, to solve the roommate problem we propose matchings that lie at t he intersection of the maximum irreversible matchings and maximum stable matchings , which are called Q-stable matchings. These matchings are core consistent and we offer an efficient algorithm for computing one of them. The outcome of the algorithm belongs to an ab sorbing set

Unveiling Hidden Values of Optimization Models with Metaheuristic Approach

Considering that the decision making process for constrained optimization problem is based on modeling, there is always room for alternative solutions because there is usually a gap between the model and the real problem it depicts. This study looks into the problem of finding such alternative solutions, the non-optimal solutions of interest for constrained optimization models, the SoI problem. SoI problems subsume finding feasible solutions of interest (FoIs) and infeasible solutions of interest (IoIs). In all cases, the interest addressed is post-solution analysis in one form or another. Post-solution analysis of a constrained optimization model occurs after the model has been solved and a good or optimal solution for it has been found. At this point, sensitivity analysis and other questions of import for decision making come into play and for this purpose the SoIs can be very valuable. An evolutionary computation approach (in particular, a population-based metaheuristic) is proposed for solving the SoI problem and a systematic approach with a feasible-infeasible- two-population genetic algorithm is demonstrated. In this study, the effectiveness of the proposed approach on finding SoIs is demonstrated with generalized assignment problems and generalized quadratic assignment problems. Also, the applications of the proposed approach on the multi-objective optimization and robust-optimization issues are examined and illustrated with two-sided matching problems and flowshop scheduling problems respectively

Essays on the partnership problem

Since the seminal Gale and Shapley (1962) paper, the field of matching theory has seen a lot of research on theoretical and empirical topics by economist, mathematicians and computer scientist alike. This dissertation aims to shed light on a relatively little discussed matching problem, the partnership problem, which is a very general type of matching problem that allows for a wide range of preference structures. In its current form, as it is used in this dissertation, it was first introduced by Fleiner (2010). There are two main contributions of this dissertation to the current state of the literature. First, in the second chapter, we look at an application of a partnership problem, a network formation model with a heterogeneous cost function. We show the existence of a unique stable outcome for this application and give an illustration of its relevance in predicting real life matching out- comes. Besides contributing to the matching theory literature, it also contributes to the network formation literature by introducing a cost function which allows for a high degree of heterogeneity. Second, in the third and fourth chapter, we consider known results from simpler matching problems and extend them to the partnership problem. As such, the main take away from these last two chapters is that, while the partnership problem is a more complicated problem with a much richer preference structure, structurally it is still very similar to the more simple matching problems. In essence, we are dealing with complex matching problems which have a surprisingly simple basic structure