Dimitrios Magos

Pareto optimality in many-to-many matching problems

Consider a many-to-many matching market that involves two finite disjoint sets, a set A of applicants and a set C of courses. Each applicant has preferences on the different sets of courses she can attend, while each course has a quota of applicants that it can admit. In this paper, we examine Pareto optimal matchings (briefly POM) in the context of such markets, that can also incorporate additional constraints, e.g., each course bearing some cost and each applicant having a limited budget available. We provide necessary and sufficient conditions for a many-to-many matching to be Pareto optimal and show that checking whether a given matching is Pareto optimal can be accomplished in O ( ? A ? 2 ? ? C ? 2 ) time. Moreover, we provide a generalized version of serial dictatorship, which can be used to obtain any many-to-many POM. We also study some structural questions related to POM. We show that, unlike in the one-to-one case, finding a maximum cardinality POM is NP-hard for many-to-many markets.

Finding All Stable Pairs and Solutions to the Many-to-Many Stable Matching Problem

The many-to-many stable matching problem (MM), defined in the context of a job market, asks for an assignment of workers to firms satisfying the quota of each agent and being stable, pairwise or setwise, with respect to given preference lists or relations. In this paper, we propose a time-optimal algorithm that identifies all stable worker--firm pairs and all stable assignments under pairwise stability, individual preferences, and the max-min criterion. We revisit the poset graph of rotations to obtain an optimal algorithm for enumerating all solutions to the MM and an improved algorithm finding the minimum-weight one. Furthermore, we establish the applicability of all aforementioned algorithms under more complex preference and stability criteria. In a constraint programming context, we introduce a constraint that models the MM and an encoding of the MM as a constraint satisfaction problem. Finally, we provide a series of computational results, including the case where side constraints are imposed.

A Branch & Cut algorithm for a four-index assignment problem

In this paper, we examine the orthogonal Latin squares (OLS) problem from an integer programming perspective. The OLS problem has a long history and its significance arises from both theoretical aspects and practical applications. The problem is formulated as a four-index assignment problem whose solutions correspond to OLS. This relationship is exploited by various routines (preliminary variable fixing, branching, etc) of the Branch & Cut algorithm we present. Clique, odd-hole and antiweb inequalities implement the ‘Cut’ component of the algorithm. For each cut type a polynomial-time separation algorithm is implemented. Extensive computational analysis examines multiple aspects concerning the design of our algorithm. The results illustrate clearly the improvement achieved over simple Branch & Bound.

Polyhedral Aspects of Stable Marriage

In the setting of the stable matching (SM) problem, it has been observed that some of the man-woman pairs cannot be removed although they participate in no stable matching, since such a removal would alter the set of solutions. These pairs are yet to be identified. Likewise (and despite the sizeable literature), some of the fundamental characteristics of the SM polytope (e.g., its dimension, its facets, etc.) have not been established. In the current work, we show that these two seemingly distant open issues are closely related. More specifically, we identify the pairs with the above-mentioned property and present a polynomial algorithm for producing a set of minimal preference lists. We utilize this result in the context of two different representations of the SM structure (rotation-poset graph and algebraic formulation) and derive the dimension of the SM polytope to obtain all alternative minimal linear descriptions.

Pareto Optimal Matchings in Many-to-Many Markets with Ties

We consider Pareto optimal matchings (POMs) in a many-to-many market of applicants and courses where applicants have preferences, which may include ties, over individual courses and lexicographic preferences over sets of courses. Since this is the most general setting examined so far in the literature, our work unifies and generalizes several known results. Specifically, we characterize POMs and introduce the Generalized Serial Dictatorship Mechanism with Ties (GSDT) that effectively handles ties via properties of network flows. We show that GSDT can generate all POMs using different priority orderings over the applicants, but it satisfies truthfulness only for certain such orderings. This shortcoming is not specific to our mechanism; we show that any mechanism generating all POMs in our setting is prone to strategic manipulation. This is in contrast to the one-to-one case (with or without ties), for which truthful mechanisms generating all POMs do exist.

The constraint of difference and total dual integrality

One of the most important logic constraints is the constraint of difference. It is imposed on a set of discrete variables requiring that they receive pairwise distinct values. This construct, initially studied in the field of Artificial Intelligence (in particular, Constraint Programming), has numerous applications and important theoretical properties. In the current work, we show that the polytope associated with this constraint is a generalized polymatroid and thus totally dual integral. As a consequence the problem of optimizing a linear function when variables are restricted to take pairwise distinct values belongs to P. Furthermore, we prove that the above problem can be solved by the greedy algorithm in O(|J| · log|J|) steps where J denotes the set indexing the variables (to receive pairwise distinct values). We establish that the dual of the above problem can also be solved in the same number of steps.

Blockers and antiblockers of stable matchings

An implicit linear description of the stable matching polytope is provided in terms of the blocker and antiblocker sets of constraints of the matroid-kernel polytope. The explicit identification of both these sets is based on a partition of the stable pairs in which each agent participates. Here, we expose the relation of such a partition to rotations. We provide a time-optimal algorithm for obtaining such a partition and establish some new related results; most importantly, that this partition is unique.

FROM ONE STABLE MARRIAGE TO THE NEXT: HOW LONG IS THE WAY? ∗

The diameter of the stable matching (stable marriage) polytope is bounded from above by

A characterization of odd-hole inequalities related to Latin squares

\left\lfloor\frac{n}{2}\right\rfloor

The stable b-matching polytope revisited

, where