Jointly Stable Matchings

In the stable marriage problem, we are given a set of men, a set of women, and each person\u27s preference list. Our task is to find a stable matching, that is, a matching admitting no unmatched (man, woman)-pair each of which improves the situation by being matched together. It is known that any instance admits at least one stable matching. In this paper, we consider a natural extension where k (>= 2) sets of preference lists L_i (1 <= i <= k) over the same set of people are given, and the aim is to find a jointly stable matching, a matching that is stable with respect to all L_i. We show that the decision problem is NP-complete already for k=2, even if each person\u27s preference list is of length at most four, while it is solvable in linear time for any k if each man\u27s preference list is of length at most two (women\u27s lists can be of unbounded length). We also show that if each woman\u27s preference lists are same in all L_i, then the problem can be solved in linear time

Smith and Rawls Share a Room

We consider one-to-one matching (roommate) problems in which agents (students) can either be matched as pairs or remain single. The aim of this paper is twofold. First, we review a key result for roommate problems (the ``lonely wolf'' theorem) for which we provide a concise and elementary proof. Second, and related to the title of this paper, we show how the often incompatible concepts of stability (represented by the political economist Adam Smith) and fairness (represented by the political philosopher John Rawls) can be reconciled for roommate problems.roommate problem, stability, fairness

Stable Marriage with Multi-Modal Preferences

We introduce a generalized version of the famous Stable Marriage problem, now based on multi-modal preference lists. The central twist herein is to allow each agent to rank its potentially matching counterparts based on more than one "evaluation mode" (e.g., more than one criterion); thus, each agent is equipped with multiple preference lists, each ranking the counterparts in a possibly different way. We introduce and study three natural concepts of stability, investigate their mutual relations and focus on computational complexity aspects with respect to computing stable matchings in these new scenarios. Mostly encountering computational hardness (NP-hardness), we can also spot few islands of tractability and make a surprising connection to the \textsc{Graph Isomorphism} problem

Smith and Rawls share a room

We consider one-to-one matching (roommate) problems in which agents (students) can either be matched as pairs or remain single. The aim of this paper is twofold. First, we review a key result for roommate problems (the ''lonely wolf'' theorem) for which we provide a concise and elementary proof. Second, and related to the title of this paper, we show how the often incompatible concepts of stability (represented by the political economist Adam Smith) and fairness (represented by the political philosopher John Rawls) can be reconciled for roommate problems

Classified Stable Matching

We introduce the {\sc classified stable matching} problem, a problem motivated by academic hiring. Suppose that a number of institutes are hiring faculty members from a pool of applicants. Both institutes and applicants have preferences over the other side. An institute classifies the applicants based on their research areas (or any other criterion), and, for each class, it sets a lower bound and an upper bound on the number of applicants it would hire in that class. The objective is to find a stable matching from which no group of participants has reason to deviate. Moreover, the matching should respect the upper/lower bounds of the classes. In the first part of the paper, we study classified stable matching problems whose classifications belong to a fixed set of ``order types.'' We show that if the set consists entirely of downward forests, there is a polynomial-time algorithm; otherwise, it is NP-complete to decide the existence of a stable matching. In the second part, we investigate the problem using a polyhedral approach. Suppose that all classifications are laminar families and there is no lower bound. We propose a set of linear inequalities to describe stable matching polytope and prove that it is integral. This integrality allows us to find various optimal stable matchings using Ellipsoid algorithm. A further ramification of our result is the description of the stable matching polytope for the many-to-many (unclassified) stable matching problem. This answers an open question posed by Sethuraman, Teo and Qian

Smith and Rawls share a room

We consider one-to-one matching (roommate) problems in which agents (students) can either be matched as pairs or remain single. The aim of this paper is twofold. First, we review a key result for roommate problems (the ``lonely wolf'' theorem) for which we provide a concise and elementary proof. Second, and related to the title of this paper, we show how the often incompatible concepts of stability (represented by the political economist Adam Smith) and fairness (represented by the political philosopher John Rawls) can be reconciled for roommate problems.F. Klijn's research was supported through the Spanish Plan Nacional I+D+I (SEJ2005- 01690) and the Generalitat de Catalunya (SGR2005-00626 and the Barcelona Economics Program of XREA)

Cell Modeling

The Air Force is currently developing new products that incorporate a variety of chemicals which may come in contact with product users. To define which chemicals are dangerous to the user, toxicity studies have been performed. However, analysis of toxicity ultimately requires models of the exposed cellular systems. This thesis provides an introduction of how to model and analyze small and large cellular systems. Understanding the underlying behavior of small models and their relation to large systems will lead to a better understanding of how the Air Force should construct intracellular models to assist in future toxicology studies. Developing analysis techniques to include steady state analysis through linearization, and then considering small reaction systems, such as the Brusselator and Schnackenberg models, led to a basic understanding of model behavior. This knowledge was applied to create new models in an effort to begin a transition from previously created models to the construction of models unique to the Air Force. Sensitivity analyses performed on existing systems furthered research efforts by developing knowledge of how systems behave under various initial conditions and perturbations of uncertain constant parameters. Analysis displayed great sensitivity within some models. This analysis was applied to a new model to look for interesting behavior such as oscillatory convergence. The new model was then incorporated into a larger model to determine how its behavior changed with respect to changes in the larger model. This knowledge of how small systems behave in relation to larger systems should help the Air Force to develop and analyze intracellular toxicology models