arXiv subjects

Yuri Faenza

Publications and source records attributed to Yuri Faenza.

At least 19 recordsLinked to original sources

arXiv2026

Fair Online Resource Allocation

We study the problem of fair online resource allocation, motivated by applications such as refugee resettlement and airline scheduling, where agents arrive sequentially and must be assigned to facilities with limited capacities. We introduce a model that maximizes the overall welfare subject to resource constraints and a Lipschitz fairness requirement, which ensures that similar agents arriving in the same batch receive similar expected outcomes. We first analyze the offline problem, proving that the value of the optimal fair allocation is at least an $\Omega(1/\gamma)$ fraction of the optimal unfair allocation, where $\gamma$ is the fairness coefficient, thereby bounding the price of fairness. For the online setting, we propose an algorithm based on dual mirror descent that enforces fairness constraints within batches while estimating optimal dual variables. We prove that this algorithm achieves sublinear regret relative to the optimal offline fluid benchmark. Finally, we validate our theoretical results using real-world data from the Refugee Economies Programme, demonstrating the algorithm's performance and examining the trade-offs between welfare maximization and fairness enforcement.

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arXiv2026

All finite lattices are stable matching lattices

We show that all finite lattices, including non-distributive lattices, arise as stable matching lattices when all agents have path-independent choice functions. This result answers an open question of Blair~\cite{blair1988lattice}. In the process, we introduce new tools to reason on general lattices for optimization purposes: the \emph{partial representation} of a lattice, which partially extends Birkhoff's representation theorem to non-distributive lattices; the \emph{distributive closure} of a lattice, which gives such a partial representation; and \emph{join constraints}, which can be added to the distributive closure to obtain a representation for the original lattice. Then, we use these techniques to show that the minimum cost stable matching problem under the same standard assumptions on choice functions is NP-hard, by establishing a connection with antimatroid theory.

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arXiv2025

Linear Programming Hierarchies Collapse under Symmetry

The presence of symmetries is one of the central structural features that make some integer programs challenging for state-of-the-art solvers. In this work, we study the efficacy of Linear Programming (LP) hierarchies in the presence of symmetries. Our main theorem unveils a connection between the algebraic structure of these relaxations and the geometry of the initial integer-empty polytope: We show that under $(k+1)$-transitive symmetries--a measure of the underlying symmetry in the problem--the corresponding relaxation at level $k$ of the hierarchy is non-empty if and only if the initial polytope intersects all $(n-k)$-dimensional faces of the hypercube. In particular, the hierarchies of Sherali-Adams, Lovász-Schrijver, and the Lift-and-Project closure are equally effective at detecting integer emptiness. Our result provides a unifying, group-theoretic characterization of the poor performance of LP-based hierarchies, and offers a simple procedure for proving lower bounds on the integrality gaps of symmetric polytopes under these hierarchies.

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arXiv2025

Feature Selection in Nonlinear SVMs via Local Search and Submodular Optimization

Embedded feature selection is a classical approach to interpretable machine learning, aiming to iden- tify the most informative variables while simultaneously training the predictive model. We address the problem of enforcing an exact limit on the number of features used in nonlinear Support Vector Machines (SVMs) by means of a hard cardinality constraint. Whereas existing hard-constraint formulations apply only to the primal formulation of linear SVMs, we embed the cardinality constraint directly into the dual formulation of a nonlinear SVM. This ensures strict control over the number of selected features while still leveraging kernelization. We model the problem as a mixed-integer nonlinear program (MINLP). Our first contribution is a local-search metaheuristic applicable to any nonlinear kernel, which exploits the decomposable structure of the formulation to efficiently explore the binary decision space. Our second and main contribution is a decomposition framework that alternates optimization over continuous and binary variables. For polynomial kernels, we reformulate the binary subproblem as the maximization of a monotone submodular function under a cardinality constraint, enabling the integration of scalable greedy algorithms into the alternating scheme. Computational experiments on benchmark datasets show that the proposed algorithms significantly outperform existing MINLP solvers and standard heuristics, delivering higher-quality solutions within practical runtimes.

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arXiv2025

Longer Lists Yield Better Matchings

Many centralized mechanisms for two-sided matching markets that enjoy strong theoretical properties assume that the planner solicits full information on the preferences of each participating agent. In particular, they expect that participants compile and communicate their complete preference lists over agents from the other side of the market. However, real-world markets are often very large and agents cannot always be expected to even produce a ranking of all options on the other side. It is therefore important to understand the impact of incomplete or truncated lists on the quality of the resultant matching. In this paper, we focus on the Serial Dictatorship mechanism in a model where each agent of the proposing side (students) has a random preference list of length $d$, sampled independently and uniformly at random from $n$ schools, each of which has one seat. Our main result shows that if the students primarily care about being matched to any school of their list (as opposed to ending up unmatched), then all students in position $i\leq n$ will prefer markets with longer lists, when $n$ is large enough. Schools on the other hand will always prefer longer lists in our model. We moreover investigate the impact of $d$ on the rank of the school that a student gets matched to. Our main result suggests that markets that are well-approximated by our hypothesis and where the demand of schools does not exceed supply should be designed with preference lists as long as reasonable, since longer lists would favor all agents.

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arXiv2025

Minimum Cut Representability of Stable Matching Problems

We introduce and study Minimum Cut Representability, a framework to solve optimization and feasibility problems over stable matchings by representing them as minimum s-t cut problems on digraphs over rotations. We provide necessary and sufficient conditions on objective functions and feasibility sets for problems to be minimum cut representable. In particular, we define the concepts of first and second order differentials of a function over stable matchings and show that a problem is minimum cut representable if and only if, roughly speaking, the objective function can be expressed solely using these differentials, and the feasibility set is a sublattice of the stable matching lattice. To demonstrate the practical relevance of our framework, we study a range of real-world applications, including problems involving school choice with siblings and a two-stage stochastic stable matching problem. We show how our framework can be used to help solving these problems.

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arXiv2025

The Power of Greedy for Online Minimum Cost Matching on the Line

We consider the online minimum cost matching problem on the line, in which there are $n$ servers and, at each of $n$ time steps, a request arrives and must be irrevocably matched to a server that has not yet been matched to, with the goal of minimizing the sum of the distances between the matched pairs. Despite achieving a worst-case competitive ratio that is exponential in $n$, the simple greedy algorithm, which matches each request to its nearest available free server, performs very well in practice. A major question is thus to explain greedy's strong empirical performance. In this paper, we aim to understand the performance of greedy over instances that are at least partially random. When both the requests and the servers are drawn uniformly and independently from $[0,1]$, we show that greedy is constant competitive, which improves over the previously best-known $O(\sqrt{n})$ bound. We extend this constant competitive ratio to a setting with a linear excess of servers, which improves over the previously best-known $O(\log^3{n})$ bound. We moreover show that in the semi-random model where the requests are still drawn uniformly and independently but where the servers are chosen adversarially, greedy achieves an $O(\log{n})$ competitive ratio. When the requests arrive in a random order but are chosen adversarially, it was previously known that greedy is $O(n)$-competitive. Even though this one-sided randomness allows a large improvement in greedy's competitive ratio compared to the model where requests are adversarial and arrive in a random order, we show that it is not sufficient to obtain a constant competitive ratio by giving a tight $Ω(\log{n})$ lower bound. These results invite further investigation about how much randomness is necessary and sufficient to obtain strong theoretical guarantees for the greedy algorithm for online minimum cost matching, on the line and beyond.

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arXiv2024

Scarf's Algorithm on Arborescence Hypergraphs

Scarf's algorithm--a pivoting procedure that finds a dominating extreme point in a down-monotone polytope--can be used to show the existence of a fractional stable matching in hypergraphs. The problem of finding a fractional stable matching in a hypergraph, however, is PPAD-complete. In this work, we study the behavior of Scarf's algorithm on arborescence hypergraphs, the family of hypergraphs in which hyperedges correspond to the paths of an arborescence. For arborescence hypergraphs, we prove that Scarf's algorithm can be implemented to find an integral stable matching in polynomial time. En route to our result, we uncover novel structural properties of bases and pivots for the more general family of network hypergraphs. Our work provides the first proof of polynomial-time convergence of Scarf's algorithm on hypergraphic stable matching problems, giving hope to the possibility of polynomial-time convergence of Scarf's algorithm for other families of polytope.

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arXiv2024

An Algorithm for the Assignment Game Beyond Additive Valuations

The assignment game, introduced by Shapley and Shubik (1971), is a classic model for two-sided matching markets between buyers and sellers. In the original assignment game, it is assumed that payments lead to transferable utility and that buyers have unit-demand valuations for items. Two important and mostly independent lines of work have studied more general settings with imperfectly transferable utility and gross substitutes valuations. Multiple efficient algorithms have been proposed for computing a competitive equilibrium, the standard solution concept in assignment games, in these two settings. Our main result is an efficient algorithm for computing competitive equilibria in a setting with both imperfectly transferable utility and gross substitutes valuations. Our algorithm combines augmenting tree techniques from maximum matching and algorithms for matroid intersection. We also show that, in two mild generalizations of our model, computing a competitive equilibrium is NP-hard.

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arXiv2023

The Total Matching Polytope of Complete Bipartite Graphs

The total matching polytope generalizes the stable set polytope and the matching polytope. In this paper, we first propose new facet-defining inequalities for the total matching polytope. We then give an exponential-sized, non-redundant description in the original space and a compact description in an extended space of the total matching polytope of complete bipartite graphs.

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arXiv2023

Discovering Opportunities in New York City's Discovery Program: Disadvantaged Students in Highly Competitive Markets

Discovery program (DISC) is a policy used by the New York City Department of Education (NYC DOE) to increase the number of admissions of students from low socio-economic background to specialized high schools. This policy has been instrumental in increasing the number of disadvantaged students attending these schools. However, assuming that students care more about the school they are assigned to rather than the type of seat they occupy (\emph{school-over-seat hypothesis}), our empirical analysis using data from 12 recent academic years shows that DISC creates about 950 in-group blocking pairs each year amongst disadvantaged students, impacting about 650 disadvantaged students every year. Moreover, we find that this program does not respect improvements, thus unintentionally creating an incentive to under-perform. These experimental results are confirmed by our theoretical analysis. In order to alleviate the concerns caused by DISC, we explore two alternative policies: the minority reserve (MR) and the joint-seat allocation (JSA) mechanisms. As our main theoretical contribution, we introduce a feature of markets, that we term high competitiveness, and we show that under this condition, JSA dominates MR for all disadvantaged students. We give sufficient conditions under which high competitiveness is verified. Data from NYC DOE satisfies the high competitiveness condition, and for this dataset our empirical results corroborate our theoretical predictions, showing the superiority of JSA. Given that JSA can be implemented by a simple modification of the classical deferred acceptance algorithm with responsive preference lists, we believe that, when the school-over-seat hypothesis holds, the discovery program can be changed for the better by implementing the JSA mechanism, leading in particular to aligned incentives for the top-performing disadvantaged students.

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arXiv2023

The Incremental Knapsack Problem with Monotone Submodular All-or-Nothing Profits

We study incremental knapsack problems with profits given by a special class of monotone submodular functions, that we dub all-or-nothing. We show that these problems are not harder to approximate than a less general class of modular incremental knapsack problems, that have been investigated in the literature. We also show that certain extensions to more general submodular functions are APX-hard.

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arXiv2023

Scarf's algorithm and stable marriages

Scarf's algorithm gives a pivoting procedure to find a special vertex -- a dominating vertex -- in down-monotone polytopes. This paper studies the behavior of Scarf's algorithm when employed to find stable matchings in bipartite graphs. First, it proves that Scarf's algorithm can be implemented to run in polynomial time, showing the first positive result on its runtime in significant settings. Second, it shows an infinite family of instances where, no matter the pivoting rule and runtime, Scarf's algorithm outputs a matching from an exponentially small subset of all stable matchings, thus showing a structural weakness of the approach.

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arXiv2023

Internal Closedness and von Neumann-Morgenstern Stability in Matching Theory: Structures and Complexity

Let $G$ be a graph and suppose we are given, for each $v \in V(G)$, a strict ordering of the neighbors of $v$. A set of matchings ${\cal M}$ of $G$ is called internally stable if there are no matchings $M,M' \in {\cal M}$ such that an edge of $M$ blocks $M'$. The sets of stable (à la Gale and Shapley) matchings and of von Neumann-Morgenstern stable matchings are examples of internally stable sets of matching. In this paper, we study, in both the marriage and the roommate case, inclusionwise maximal internally stable sets of matchings. We call those sets internally closed. By building on known and newly developed algebraic structures associated to sets of matchings, we investigate the complexity of deciding if a set of matchings is internally closed or von Neumann-Morgenstern stable, and of finding sets with those properties.

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arXiv2021

Slack matrices, $k$-products, and $2$-level polytopes

In this paper, we study algorithmic questions concerning products of matrices and their consequences for recognition algorithms for polyhedra. The 1-product of matrices $S_1$, $S_2$ is a matrix whose columns are the concatenation of each column of $S_1$ with each column of $S_2$. The $k$-product generalizes the $1$-product, by taking as input two matrices $S_1, S_2$ together with $k-1$ special rows of each of those matrices, and outputting a certain composition of $S_1,S_2$. Our study is motivated by a close link between the 1-product of matrices and the Cartesian product of polytopes, and more generally between the $k$-product of matrices and the glued product of polytopes. These connections rely on the concept of slack matrix, which gives an algebraic representation of classes of affinely equivalent polytopes. The slack matrix recognition problem is the problem of determining whether a given matrix is a slack matrix. This is an intriguing problem whose complexity is unknown. Our algorithm reduces the problem to instances which cannot be expressed as $k$-products of smaller matrices. In the second part of the paper, we give a combinatorial interpretation of $k$-products for two well-known classes of polytopes: 2-level matroid base polytopes and stable set polytopes of perfect graphs. We also show that the slack matrix recognition problem is polynomial-time solvable for such polytopes. Those two classes are special cases of $2$-level polytopes, for which we conjecture that the slack matrix recognition problem is polynomial-time solvable.

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arXiv2021

Understanding popular matchings via stable matchings

Let $G = (A \cup B, E)$ be an instance of the stable marriage problem with strict preference lists. A matching $M$ is popular in $G$ if $M$ does not lose a head-to-head election against any matching where vertices are voters. Every stable matching is a min-size popular matching; another subclass of popular matchings that always exist and can be easily computed is the set of dominant matchings. A popular matching $M$ is dominant if $M$ wins the head-to-head election against any larger matching. Thus every dominant matching is a max-size popular matching and it is known that the set of dominant matchings is the linear image of the set of stable matchings in an auxiliary graph. Results from the literature seem to suggest that stable and dominant matchings behave, from a complexity theory point of view, in a very similar manner within the class of popular matchings. The goal of this paper is to show that indeed there are differences in the tractability of stable and dominant matchings, and to investigate further their importance for popular matchings. First, we show that it is easy to check if all popular matchings are also stable, however it is co-NP hard to check if all popular matchings are also dominant. Second, we show how some new and recent hardness results on popular matching problems can be deduced from the NP-hardness of certain problems on stable matchings, also studied in this paper, thus showing that stable matchings can be employed not only to show positive results on popular matching (as is known), but also most negative ones. Problems for which we show new hardness results include finding a min-size (resp. max-size) popular matching that is not stable (resp. dominant). A known result for which we give a new and simple proof is the NP-hardness of finding a popular matching when $G$ is non-bipartite.

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arXiv2021

Affinely representable lattices, stable matchings, and choice functions

Birkhoff's representation theorem (Birkhoff, 1937) defines a bijection between elements of a distributive lattice and the family of upper sets of an associated poset. Although not used explicitly, this result is at the backbone of the combinatorial algorithm by Irving et al. (1987) for maximizing a linear function over the set of stable matchings in Gale and Shapley's stable marriage model (Gale and Shapley, 1962). In this paper, 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. To the best of our knowledge, this model generalizes all models from the literature 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.

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arXiv2020

Approximation Algorithms for The Generalized Incremental Knapsack Problem

We introduce and study a discrete multi-period extension of the classical knapsack problem, dubbed generalized incremental knapsack. In this setting, we are given a set of $n$ items, each associated with a non-negative weight, and $T$ time periods with non-decreasing capacities $W_1 \leq \dots \leq W_T$. When item $i$ is inserted at time $t$, we gain a profit of $p_{it}$; however, this item remains in the knapsack for all subsequent periods. The goal is to decide if and when to insert each item, subject to the time-dependent capacity constraints, with the objective of maximizing our total profit. Interestingly, this setting subsumes as special cases a number of recently-studied incremental knapsack problems, all known to be strongly NP-hard. Our first contribution comes in the form of a polynomial-time $(\frac{1}{2}-ε)$-approximation for the generalized incremental knapsack problem. This result is based on a reformulation as a single-machine sequencing problem, which is addressed by blending dynamic programming techniques and the classical Shmoys-Tardos algorithm for the generalized assignment problem. Combined with further enumeration-based self-reinforcing ideas and newly-revealed structural properties of nearly-optimal solutions, we turn our basic algorithm into a quasi-polynomial time approximation scheme (QPTAS). Hence, under widely believed complexity assumptions, this finding rules out the possibility that generalized incremental knapsack is APX-hard.

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