Two-sided matching markets are a cornerstone of modern economics. They model a wide range of applications such as ride-sharing, online dating, job positioning, school admissions, and many more. In many of those markets, monetary exchange does not play a role. For instance, the New York City public high school system is free of charge. Thus, the decision on how eighth-graders are assigned to public high schools must be made using concepts of fairness rather than price. There has been therefore a huge amount of literature, mostly in the economics community, defining various concepts of fairness in different settings and showing the existence of matchings that satisfy these fairness conditions. Those concepts have enjoyed wide-spread success, inside and outside academia. However, finding such matchings is as important as showing their existence. Moreover, it is crucial to have fast (i.e., polynomial-time) algorithms as the size of the markets grows. In many cases, modern algorithmic tools must be employed to tackle the intractability issues arising from the big data era.

The aim of my research is to provide mathematically rigorous and provably fast algorithms to find solutions that extend and improve over a well-studied concept of fairness in two-sided markets known as stability. This concept was initially employed by the National Resident Matching Program in assigning medical doctors to hospitals, and is now widely used, for instance, by cities in the US for assigning students to public high schools and by certain refugee agencies to relocate asylum seekers. In the classical model, a stable matching can be found efficiently using the renowned deferred acceptance algorithm by Gale and Shapley. However, stability by itself does not take care of important concerns that arose recently, some of which were featured in national newspapers. Some examples are: how can we make sure students get admitted to the best school they deserve, and how can we enforce diversity in a cohort of students? By building on known and new tools from Mathematical Programming, Combinatorial Optimization, and Order Theory, my goal is to provide fast algorithms to answer questions like those above, and test them on real-world data.

In Chapter 1, I introduce the stable matching problem and related concepts, as well as its applications in different markets. In Chapter 2, we investigate two extensions introduced in the framework of school choice that aim at finding an assignment that is more favorable to students -- legal assignments and the Efficiency Adjusted Deferred Acceptance Mechanism (EADAM) -- through the lens of classical theory of stable matchings. We prove that the set of legal assignments is exactly the set of stable assignments in another instance. Our result implies that essentially all optimization problems over the set of legal assignments can be solved within the same time bound needed for solving it over the set of stable assignments. We also give an algorithm that obtains the assignment output of EADAM. Our algorithm has the same running time as that of the deferred acceptance algorithm, hence largely improving in both theory and practice over known algorithms. In Chapter 3, we introduce a property of distributive lattices, which we term as affine representability, and show its role in efficiently solving linear optimization problems over the elements of a distributive lattice, as well as describing the convex hull of the characteristic vectors of the lattice elements.

We apply this concept to the stable matching model with path-independent quota-filling choice functions, thus giving efficient algorithms and a compact polyhedral description for this model. Such choice functions can be used to model many complex real-world decision rules that are not captured by the classical model, such as those with diversity concerns. To the best of our knowledge, this model generalizes all those for which similar results were known, and our paper is the first that proposes efficient algorithms for stable matchings with choice functions, beyond classical extensions of the Deferred Acceptance algorithm. In Chapter 4, we study the discovery program (DISC), which is an affirmative action policy used by the New York City Department of Education (NYC DOE) for specialized high schools; and explore two other affirmative action policies that can be used to minimally modify and improve the discovery program: the minority reserve (MR) and the joint-seat allocation (JSA) mechanism.

Although the discovery program is beneficial in increasing the number of admissions for disadvantaged students, our empirical analysis of the student-school matches from the 12 recent academic years (2005-06 to 2016-17) shows that about 950 in-group blocking pairs were created each year amongst disadvantaged group of students, impacting about 650 disadvantaged students every year. Moreover, we find that this program usually benefits lower-performing disadvantaged students more than top-performing disadvantaged students (in terms of the ranking of their assigned schools), thus unintentionally creating an incentive to under-perform. On the contrary, we show, theoretically by employing choice functions, that (i) both MR and JSA result in no in-group blocking pairs, and (ii) JSA is weakly group strategy-proof, ensures that at least one disadvantaged is not worse off, and when reservation quotas are carefully chosen then no disadvantaged student is worse-off. We show that each of these properties is not satisfied by DISC.

In the general setting, we show that there is no clear winner in terms of the matchings provided by DISC, JSA, and MR, from the perspective of disadvantaged students. We however characterize a condition for markets, that we term high competitiveness, where JSA dominates MR for disadvantaged students. This condition is verified, in particular, in certain markets when there is a higher demand for seats than supply, and the performances of disadvantaged students are significantly lower than that of advantaged students. Data from NYC DOE satisfy the high competitiveness condition, and for this dataset our empirical results corroborate our theoretical predictions, showing the superiority of JSA. We believe that the discovery program, and more generally affirmative action mechanisms, can be changed for the better by implementing the JSA mechanism, leading to incentives for the top-performing disadvantaged students while providing many benefits of the affirmative action program.