arXiv subjects

Alfredo Torrico

Publications and source records attributed to Alfredo Torrico.

14 recordsLinked to original sources

arXiv2026

Two-sided Assortment Optimization: Adaptivity Gaps and Approximation Algorithms

To address efficiency and design challenges in choice-based matching platforms, we introduce a two-sided assortment optimization framework under general choice preferences. The goal in this problem is to maximize the expected number of matches by deciding which assortments are displayed to the agents and the order in which they are shown. In this context, we identify several classes of policies that platforms can use in their design. Our goals are: (1) to measure the value that one class of policies has over another one, and (2) to approximately solve the optimization problem itself for a given class. For (1), we define the adaptivity gap as the worst-case ratio between the optimal values of two different policy classes. First, we show that the gap between the class of policies that statically show assortments to one-side first and the class of policies that adaptively show assortments to one-side first is exactly $e/(e-1)$. Second, we show that the gap between the latter class of policies and the fully adaptive class of policies that show assortments to agents one by one is exactly $2$. We also note that the worst policies are those who simultaneously show assortments to all the agents. For (2), we first design a polynomial time algorithm that achieves a $1/4$ approximation factor within the class of policies that adaptively show assortments to agents one by one. Furthermore, when agents' preferences are governed by multinomial-logit models, we show that a 0.067 approximation factor can be obtained within the class of policies that show assortments to all agents at once. We further generalize our results to constrained assortment settings, where we impose an upper bound on the size of the displayed assortments. Finally, we present a computational study to evaluate the empirical performance of our theoretical guarantees.

math.OC↗

arXiv2026

Equitable Congestion Pricing under the Markovian Traffic Model: An Application to Bogota

Congestion pricing is used to raise revenues and reduce traffic and pollution. However, people have heterogeneous spatial demand patterns and willingness (or ability) to pay tolls, and so pricing may have substantial equity implications. We develop a data-driven approach to design congestion pricing given policymakers' equity and efficiency objectives. First, algorithmically, we extend the Markovian traffic equilibrium setting introduced by Baillon & Cominetti (2008) to model heterogeneous populations and incorporate prices and outside options such as public transit. In this setting, we show that a unique equilibrium exists. Second, via a detailed case study, we empirically evaluate various pricing schemes using data collected by an industry partner in the city of Bogota, one of the most congested cities in the world. We find that pricing personalized to each economic stratum can be substantially more efficient and equitable than uniform pricing; however, non-personalized but area-based pricing can recover much of the gap.

cs.GT↗

arXiv2026

The Dating Heuristic: A Provably Strong Matching Algorithm for Dating Platforms

Motivated by online dating platforms, we study the problem of selecting which subset of profiles to display to each user in each period. Users observe the profiles set by the platform, decide which of them to like, and a match occurs if and only if two users mutually like each other, potentially across different periods. The platform aims to maximize the expected number of matches produced over the entire time horizon, and users' behavior -- captured by their like probabilities -- may depend on their history. We develop a general theoretical model that captures the dynamic, two-sided nature of the problem and the influence of users' past experiences on their future behavior. We focus on one-lookahead policies and propose the Integral Dating Heuristic (DH-int), providing formal performance guarantees: DH-int achieves a uniform $1-1/e$ guarantee across all platform designs under reasonable assumptions. Our empirical analysis, using proprietary data from a major U.S.-based dating app, confirms that DH-int consistently outperforms other benchmarks such as Greedy, Perfect Matching and DH, and approaches the theoretical upper bound across multiple platform designs and variants of the history effect. The superior performance of DH-int is driven primarily by its careful balancing of initial and follow-up interactions, which accounts for the two-sided nature of the market.

math.OC↗

arXiv2024

Revenue Maximization in Choice-Based Matching Markets

The primary contribution of this paper resides in devising constant-factor approximation guarantees for revenue maximization in two-sided matching markets, under general pairwise rewards. A major distinction between our work and state-of-the-art results in this context (Ashlagi et al., 2022; Torrico et al., 2023) is that, for the first time, we are able to address reward maximization, reflected by assigning each customer-supplier pair an arbitrarily-valued reward. The specific type of performance guarantees we attain depends on whether one considers the customized model or the inclusive model. The fundamental difference between these settings lies in whether the platform should display to each supplier all selecting customers, as in the inclusive model, or whether the platform can further personalize this set, as in the customized model. Technically speaking, our algorithmic approach and its analysis revolve around presenting novel linear relaxations, leveraging convex stochastic orders, employing approximate dynamic programming, and developing tailor-made analytical ideas. In both models considered, these ingredients allow us to overcome the lack of submodularity and subadditivity that stems from pairwise rewards, plaguing the applicability of existing methods.

cs.GT↗

arXiv2024

Stable Matching with Contingent Priorities

Using school choice as a motivating example, we introduce a stylized model of a many-to-one matching market where the clearinghouse aims to implement contingent priorities, i.e., priorities that depend on the current assignment, to prioritize students with siblings and match them together. We provide a series of guidelines and introduce two natural approaches to implement them: (i) absolute, whereby a prioritized student can displace any student without siblings assigned to the school, and (ii) partial, whereby prioritized students can only displace students that have a less favorable lottery than their priority provider. We study several properties of the corresponding mechanisms, including the existence of a stable assignment under contingent priorities, the complexity of deciding whether there exists one, and its incentive properties. Furthermore, we introduce a soft version of these priorities to guarantee existence, and we provide mathematical programming formulations to find such stable matching or certify that one does not exist. Finally, using data from the Chilean school choice system, we show that our framework can significantly increase the number of students assigned to their top preference and the number of siblings assigned together relative to current practice.

cs.GT↗

arXiv2024

Capacity Planning in Stable Matching

Motivated by the shortage of seats that the Chilean school choice system is facing, we introduce the problem of jointly increasing school capacities and finding a student-optimal assignment in the expanded market. Due to the theoretical and practical complexity of the problem, we provide a comprehensive set of tools to solve the problem, including different mathematical programming formulations, a cutting plane algorithm, and two heuristics that allow obtaining near-optimal solutions quickly. On the theoretical side, we show the correctness of our formulations, different properties of the objective and feasible region that facilitate computation, and also several properties of the underlying mechanism to find a student-optimal matching under capacity expansions. On the computational side, we use data from the Chilean school choice system to demonstrate the impact of our framework and derive insights that could help alleviate the problem. Our results show that each additional seat can benefit multiple students and that we can effectively target the assignment of previously unassigned students or improve the assignment of several students through improvement chains. Nevertheless, our results show that the marginal effect of each additional seat is decreasing and that simply adding seats is insufficient to ensure every student gets assigned to some school. Finally, we discuss several extensions of our framework, showcasing its flexibility to accommodate different needs.

cs.GT↗

arXiv2023

Multi-agent Assortment Optimization in Sequential Matching Markets

In this work, we study the multi-agent assortment optimization problem in the two-sided sequential matching model introduced by Ashlagi et al. (2022). The setting is the following: we (the platform) offer a menu of suppliers to each customer. Then, every customer selects, simultaneously and independently, to match with a supplier or to remain unmatched. Each supplier observes the subset of customers that selected them, and choose either to match a customer or to leave the system. Therefore, a match takes place if both a customer and a supplier sequentially select each other. Each agent's behavior is probabilistic and determined by a discrete choice model. Our goal is to choose an assortment family that maximizes the expected revenue of the matching. Given the hardness of the problem, we show a $1-1/e$-approximation factor for the heterogeneous setting where customers follow general choice models and suppliers follow a general choice model whose demand function is monotone and submodular. Our approach is flexible enough to allow for different assortment constraints and for a revenue objective function. Furthermore, we design an algorithm that beats the $1-1/e$ barrier and, in fact, is asymptotically optimal when suppliers follow the classic multinomial-logit choice model and are sufficiently selective. We finally provide other results and further insights. Notably, in the unconstrained setting where customers and suppliers follow multinomial-logit models, we design a simple and efficient approximation algorithm that appropriately randomizes over a family of nested-assortments. Also, we analyze various aspects of the matching market model that lead to several operational insights, such as the fact that matching platforms can benefit from allowing the more selective agents to initiate the matchmaking process.

cs.GT↗

arXiv2022

Capacity Variation in the Many-to-one Stable Matching

The many-to-one stable matching problem provides the fundamental abstraction of several real-world matching markets such as school choice and hospital-resident allocation. The agents on both sides are often referred to as residents and hospitals. The classical setup assumes that the agents rank the opposite side and that the capacities of the hospitals are fixed. It is known that increasing the capacity of a single hospital improves the residents' final allocation. On the other hand, reducing the capacity of a single hospital deteriorates the residents' allocation. In this work, we study the computational complexity of finding the optimal variation of hospitals' capacities that leads to the best outcome for the residents, subject to stability and a capacity variation constraint. First, we show that the decision problem of finding the optimal capacity expansion is NP-complete and the corresponding optimization problem is inapproximable within a certain factor. This result holds under strict and complete preferences, and even if we allocate extra capacities to disjoint sets of hospitals. Second, we obtain analogous computational complexity results for the problem of capacity reduction. Finally, we study the variants of these problems when the goal is to maximize the size of the final matching under incomplete preference lists.

cs.CC↗

arXiv2021

Preserving Diversity when Partitioning: A Geometric Approach

Diversity plays a crucial role in multiple contexts such as team formation, representation of minority groups and generally when allocating resources fairly. Given a community composed by individuals of different types, we study the problem of partitioning this community such that the global diversity is preserved as much as possible in each subgroup. We consider the diversity metric introduced by Simpson in his influential work that, roughly speaking, corresponds to the inverse probability that two individuals are from the same type when taken uniformly at random, with replacement, from the community of interest. We provide a novel perspective by reinterpreting this quantity in geometric terms. We characterize the instances in which the optimal partition exactly preserves the global diversity in each subgroup. When this is not possible, we provide an efficient polynomial-time algorithm that outputs an optimal partition for the problem with two types. Finally, we discuss further challenges and open questions for the problem that considers more than two types.

cs.DM↗

arXiv2020

Structured Robust Submodular Maximization: Offline and Online Algorithms

Constrained submodular function maximization has been used in subset selection problems such as selection of most informative sensor locations. While these models have been quite popular, the solutions Constrained submodular function maximization has been used in subset selection problems such as selection of most informative sensor locations. While these models have been quite popular, the solutions obtained via this approach are unstable to perturbations in data defining the submodular functions. Robust submodular maximization has been proposed as a richer model that aims to overcome this discrepancy as well as increase the modeling scope of submodular optimization. In this work, we consider robust submodular maximization with structured combinatorial constraints and give efficient algorithms with provable guarantees. Our approach is applicable to constraints defined by single or multiple matroids, knapsack as well as distributionally robust criteria. We consider both the offline setting where the data defining the problem is known in advance as well as the online setting where the input data is revealed over time. For the offline setting, we give a general (nearly) optimal bi-criteria approximation algorithm that relies on new extensions of classical algorithms for submodular maximization. For the online version of the problem, we give an algorithm that returns a bi-criteria solution with sub-linear regret.

cs.DS↗

arXiv2020

On the Unreasonable Effectiveness of the Greedy Algorithm: Greedy Adapts to Sharpness

Submodular maximization has been widely studied over the past decades, mostly because of its numerous applications in real-world problems. It is well known that the standard greedy algorithm guarantees a worst-case approximation factor of 1-1/e when maximizing a monotone submodular function under a cardinality constraint. However, empirical studies show that its performance is substantially better in practice. This raises a natural question of explaining this improved performance of the greedy algorithm. In this work, we define sharpness for submodular functions as a candidate explanation for this phenomenon. The sharpness criterion is inspired by the concept of strong convexity in convex optimization. We show that the greedy algorithm provably performs better as the sharpness of the submodular function increases. This improvement ties in closely with the faster convergence rates of first order methods for sharp functions in convex optimization. Finally, we perform a computational study to empirically support our theoretical results and show that sharpness explains the greedy performance better than other justifications in the literature.

cs.DS↗

arXiv2018

Dynamic Relaxations for Online Bipartite Matching

Online bipartite matching (OBM) is a fundamental model underpinning many important applications, including search engine advertisement, website banner and pop-up ads, and ride-hailing. We study the i.i.d. OBM problem, where one side of the bipartition is fixed and known in advance, while nodes from the other side appear sequentially as i.i.d. realizations of an underlying distribution, and must immediately be matched or discarded. We introduce dynamic relaxations of the set of achievable matching probabilities, show how they theoretically dominate lower-dimensional, static relaxations from previous work, and perform a polyhedral study to theoretically examine the new relaxations' strength. We also discuss how to derive heuristic policies from the relaxations' dual prices, in a similar fashion to dynamic resource prices used in network revenue management. We finally present a computational study to demonstrate the empirical quality of the new relaxations and policies.

math.OC↗

arXiv2018

Efficient algorithms for robust submodular maximization under matroid constraints

In this work, we consider robust submodular maximization with matroid constraints. We give an efficient bi-criteria approximation algorithm that outputs a small family of feasible sets whose union has (nearly) optimal objective value. This algorithm theoretically performs less function calls than previous works at cost of adding more elements to the final solution. We also provide significant implementation improvements showing that our algorithm outperforms the algorithms in the existing literature. We finally assess the performance of our contributions in three real-world applications.

cs.DS↗

arXiv2013

Additive consistency of risk measures and its application to risk-averse routing in networks

This paper investigates the use of risk measures and theories of choice for modeling risk-averse route choice and traffic network equilibrium with random travel times. We interpret the postulates of these theories in the context of routing, and we identify additive consistency as a plausible and relevant condition that allows to reduce risk-averse route choice to a standard shortest path problem. Within the classical theories of choice under risk, we show that the only preferences that satisfy this consistency property are the ones induced by the entropic risk measures.

math.OC↗