arXiv subjects

Amin Saberi

Publications and source records attributed to Amin Saberi.

At least 37 records · Page 2Linked to original sources

arXiv2025

Stable Matching with Interviews

In several two-sided markets, including labor and dating, agents typically have limited information about their preferences prior to mutual interactions. This issue can result in matching frictions, as arising in the labor market for medical residencies, where high application rates are followed by a large number of interviews. Yet, the extensive literature on two-sided matching primarily focuses on models where agents know their preferences, leaving the interactions necessary for preference discovery largely overlooked. This paper studies this problem using an algorithmic approach, extending Gale-Shapley's deferred acceptance to this context. Two algorithms are proposed. The first is an adaptive algorithm that expands upon Gale-Shapley's deferred acceptance by incorporating interviews between applicants and positions. Similar to deferred acceptance, one side sequentially proposes to the other. However, the order of proposals is carefully chosen to ensure an interim stable matching is found. Furthermore, with high probability, the number of interviews conducted by each applicant or position is limited to $O(\log^2 n)$. In many seasonal markets, interactions occur more simultaneously, consisting of an initial interview phase followed by a clearing stage. We present a non-adaptive algorithm for generating a single stage set of in tiered random markets. The algorithm finds an interim stable matching in such markets while assigning no more than $O(\log^3 n)$ interviews to each applicant or position.

cs.DS↗

arXiv2024

CHESS: Contextual Harnessing for Efficient SQL Synthesis

Translating natural language questions into SQL queries, known as text-to-SQL, is a long-standing research problem. Effective text-to-SQL synthesis can become very challenging due to (i) the extensive size of database catalogs (descriptions of tables and their columns) and database values, (ii) reasoning over large database schemas, (iii) ensuring the functional validity of the generated queries, and (iv) navigating the ambiguities of natural language questions. We introduce CHESS, a Large Language Model (LLM) based multi-agent framework for efficient and scalable SQL synthesis, comprising four specialized agents, each targeting one of the aforementioned challenges: the Information Retriever (IR) extracts relevant data, the Schema Selector (SS) prunes large schemas, the Candidate Generator (CG) generates high-quality candidates and refines queries iteratively, and the Unit Tester (UT) validates queries through LLM-based natural language unit tests. Our framework offers configurable features that adapt to various deployment constraints, including 1) Supporting industrial-scale databases: leveraging the Schema Selector agent, CHESS efficiently narrows down very large database schemas into manageable sub-schemas, boosting system accuracy by approximately $2\%$ and reducing the number of LLM tokens by $\times 5$. 2) State-of-the-Art privacy-preserving performance: Among the methods using open-source models, CHESS achieves state-of-the-art performance, resulting in a high-performing, privacy-preserving system suitable for industrial deployment. 3) Scalablity with additional compute budget: In settings with high computational budgets, CHESS achieves $71.10\%$ accuracy on the BIRD test set, within $2\%$ of the leading proprietary method, while requiring approximately $83\%$ fewer LLM calls.

cs.LG↗

arXiv2024

Improved Approximations for Stationary Bipartite Matching: Beyond Probabilistic Independence

We study stationary online bipartite matching, where both types of nodes--offline and online--arrive according to Poisson processes. Offline nodes wait to be matched for some random time, determined by an exponential distribution, while online nodes need to be matched immediately. This model captures scenarios such as deceased organ donation and time-sensitive task assignments, where there is an inflow of patients and workers (offline nodes) with limited patience, while organs and tasks (online nodes) must be assigned upon arrival. We present an efficient online algorithm that achieves a $(1-1/e+δ)$-approximation to the optimal online policy's reward for a constant $δ> 0$, simplifying and improving previous work by Aouad and Saritaç (2022). Our solution combines recent online matching techniques, particularly pivotal sampling, which enables correlated rounding of tighter linear programming approximations, and a greedy-like algorithm. A key technical component is the analysis of a stochastic process that exploits subtle correlations between offline nodes, using renewal theory. A byproduct of our result is an improvement to the best-known competitive ratio--that compares an algorithm's performance to the optimal offline policy--via a $(1-1/\sqrt{e} + η)$-competitive algorithm for a universal constant $η> 0$, advancing the results of Patel and Wajc (2024).

cs.DS↗

arXiv2024

CHASE-SQL: Multi-Path Reasoning and Preference Optimized Candidate Selection in Text-to-SQL

In tackling the challenges of large language model (LLM) performance for Text-to-SQL tasks, we introduce CHASE-SQL, a new framework that employs innovative strategies, using test-time compute in multi-agent modeling to improve candidate generation and selection. CHASE-SQL leverages LLMs' intrinsic knowledge to generate diverse and high-quality SQL candidates using different LLM generators with: (1) a divide-and-conquer method that decomposes complex queries into manageable sub-queries in a single LLM call; (2) chain-of-thought reasoning based on query execution plans, reflecting the steps a database engine takes during execution; and (3) a unique instance-aware synthetic example generation technique, which offers specific few-shot demonstrations tailored to test questions.To identify the best candidate, a selection agent is employed to rank the candidates through pairwise comparisons with a fine-tuned binary-candidates selection LLM. This selection approach has been demonstrated to be more robust over alternatives. The proposed generators-selector framework not only enhances the quality and diversity of SQL queries but also outperforms previous methods. Overall, our proposed CHASE-SQL achieves the state-of-the-art execution accuracy of 73.0% and 73.01% on the test set and development set of the notable BIRD Text-to-SQL dataset benchmark, rendering CHASE-SQL the top submission of the leaderboard (at the time of paper submission).

cs.LG↗

arXiv2024

Hardness of Approximate Sperner and Applications to Envy-Free Cake Cutting

Given a so called ''Sperner coloring'' of a triangulation of the $D$-dimensional simplex, Sperner's lemma guarantees the existence of a rainbow simplex, i.e. a simplex colored by all $D+1$ colors. However, finding a rainbow simplex was the first problem to be proven $\mathsf{PPAD}$-complete in Papadimitriou's classical paper introducing the class $\mathsf{PPAD}$ (1994). In this paper, we prove that the problem does not become easier if we relax ''all $D+1$ colors'' to allow some fraction of missing colors: in fact, for any constant $D$, finding even a simplex with just three colors remains $\mathsf{PPAD}$-complete! Our result has an interesting application for the envy-free cake cutting from fair division. It is known that if agents value pieces of cake using general continuous functions satisfying a simple boundary condition (''a non-empty piece is better than an empty piece of cake''), there exists an envy-free allocation with connected pieces. We show that for any constant number of agents it is $\mathsf{PPAD}$-complete to find an allocation -- even using any constant number of possibly disconnected pieces -- that makes just three agents envy-free. Our results extend to super-constant dimension, number of agents, and number of pieces, as long as they are asymptotically bounded by any $\log^{1-Ω(1)}(ε)$, where $ε$ is the precision parameter (side length for Sperner and approximate envy-free for cake cutting).

cs.CC↗

arXiv2024

New Philosopher Inequalities for Online Bayesian Matching, via Pivotal Sampling

We study the polynomial-time approximability of the optimal online stochastic bipartite matching algorithm, initiated by Papadimitriou et al. (EC'21). Here, nodes on one side of the graph are given upfront, while at each time $t$, an online node and its edge weights are drawn from a time-dependent distribution. The optimal algorithm is $\textsf{PSPACE}$-hard to approximate within some universal constant. We refer to this optimal algorithm, which requires time to think (compute), as a philosopher, and refer to polynomial-time online approximations of the above as philosopher inequalities. The best known philosopher inequality for online matching yields a $0.652$-approximation. In contrast, the best possible prophet inequality, or approximation of the optimum offline solution, is $0.5$. Our main results are a $0.678$-approximate algorithm and a $0.685$-approximation for a vertex-weighted special case. Notably, both bounds exceed the $0.666$-approximation of the offline optimum obtained by Tang, Wu, and Wu (STOC'22) for the vertex-weighted problem. Building on our algorithms and the recent black-box reduction of Banihashem et al. (SODA'24), we provide polytime (pricing-based) truthful mechanisms which $0.678$-approximate the social welfare of the optimal online allocation for bipartite matching markets. Our online allocation algorithm relies on the classic pivotal sampling algorithm (Srinivasan FOCS'01, Gandhi et al. J.ACM'06), along with careful discarding to obtain negative correlations between offline nodes. Consequently, the analysis boils down to examining the distribution of a weighted sum $X$ of negatively correlated Bernoulli variables, specifically lower bounding its mass below a threshold, $\mathbb{E}[\min(1,X)]$, of possible independent interest. Interestingly, our bound relies on an imaginary invocation of pivotal sampling.

cs.DS↗

arXiv2024

MAGNOLIA: Matching Algorithms via GNNs for Online Value-to-go Approximation

Online Bayesian bipartite matching is a central problem in digital marketplaces and exchanges, including advertising, crowdsourcing, ridesharing, and kidney exchange. We introduce a graph neural network (GNN) approach that emulates the problem's combinatorially-complex optimal online algorithm, which selects actions (e.g., which nodes to match) by computing each action's value-to-go (VTG) -- the expected weight of the final matching if the algorithm takes that action, then acts optimally in the future. We train a GNN to estimate VTG and show empirically that this GNN returns high-weight matchings across a variety of tasks. Moreover, we identify a common family of graph distributions in spatial crowdsourcing applications, such as rideshare, under which VTG can be efficiently approximated by aggregating information within local neighborhoods in the graphs. This structure matches the local behavior of GNNs, providing theoretical justification for our approach.

cs.LG↗

arXiv2024

Approximating Optimum Online for Capacitated Resource Allocation

We study online capacitated resource allocation, a natural generalization of online stochastic max-weight bipartite matching. This problem is motivated by ride-sharing and Internet advertising applications, where online arrivals may have the capacity to serve multiple offline users. Our main result is a polynomial-time online algorithm which is $(1/2 + κ)$-approximate to the optimal online algorithm for $κ= 0.0115$. This can be contrasted to the (tight) $1/2$-competitive algorithms to the optimum offline benchmark from the prophet inequality literature. Optimum online is a recently popular benchmark for online Bayesian problems which can use unbounded computation, but not "prophetic" knowledge of future inputs. Our algorithm (which also works for the case of stochastic rewards) rounds a generalized LP relaxation from the unit-capacity case via a two-proposal algorithm, as in previous works in the online matching literature. A key technical challenge in deriving our guarantee is bounding the positive correlation among users introduced when rounding our LP relaxation online. Unlike in the case of unit capacities, this positive correlation is unavoidable for guarantees beyond $1/2$. Conceptually, our results show that the study of optimum online as a benchmark can reveal problem-specific insights that are irrelevant to competitive analysis.

cs.DS↗

arXiv2024

Locality-Aware Graph-Rewiring in GNNs

Graph Neural Networks (GNNs) are popular models for machine learning on graphs that typically follow the message-passing paradigm, whereby the feature of a node is updated recursively upon aggregating information over its neighbors. While exchanging messages over the input graph endows GNNs with a strong inductive bias, it can also make GNNs susceptible to over-squashing, thereby preventing them from capturing long-range interactions in the given graph. To rectify this issue, graph rewiring techniques have been proposed as a means of improving information flow by altering the graph connectivity. In this work, we identify three desiderata for graph-rewiring: (i) reduce over-squashing, (ii) respect the locality of the graph, and (iii) preserve the sparsity of the graph. We highlight fundamental trade-offs that occur between spatial and spectral rewiring techniques; while the former often satisfy (i) and (ii) but not (iii), the latter generally satisfy (i) and (iii) at the expense of (ii). We propose a novel rewiring framework that satisfies all of (i)--(iii) through a locality-aware sequence of rewiring operations. We then discuss a specific instance of such rewiring framework and validate its effectiveness on several real-world benchmarks, showing that it either matches or significantly outperforms existing rewiring approaches.

cs.LG↗

arXiv2024

Sublinear Algorithms for TSP via Path Covers

We study sublinear time algorithms for the traveling salesman problem (TSP). First, we focus on the closely related {\em maximum path cover} problem, which asks for a collection of vertex disjoint paths that include the maximum number of edges. We show that for any fixed $ε> 0$, there is an algorithm that $(1/2 - ε)$-approximates the maximum path cover size of an $n$-vertex graph in $\widetilde{O}(n)$ time. This improves upon a $(3/8-ε)$-approximate $\widetilde{O}(n \sqrt{n})$-time algorithm of Chen, Kannan, and Khanna [ICALP'20]. Equipped with our path cover algorithm, we give an $\widetilde{O}(n)$ time algorithm that estimates the cost of $(1,2)$-TSP within a factor of $(1.5+ε)$ which is an improvement over a folklore $(1.75 + ε)$-approximate $\widetilde{O}(n)$-time algorithm, as well as a $(1.625+ε)$-approximate $\widetilde{O}(n\sqrt{n})$-time algorithm of [CHK ICALP'20]. For graphic TSP, we present an $\widetilde{O}(n)$ algorithm that estimates the cost of graphic TSP within a factor of $1.83$ which is an improvement over a $1.92$-approximate $\widetilde{O}(n)$ time algorithm due to [CHK ICALP'20, Behnezhad FOCS'21]. We show that the approximation can be further improved to $1.66$ using $n^{2-Ω(1)}$ time. All of our $\widetilde{O}(n)$ time algorithms are information-theoretically time-optimal up to poly log n factors. Additionally, we show that our approximation guarantees for path cover and $(1,2)$-TSP hit a natural barrier: We show better approximations require better sublinear time algorithms for the well-studied maximum matching problem.

cs.DS↗

arXiv2024

Linear Programming Based Near-Optimal Pricing for Laminar Bayesian Online Selection

The Bayesian online selection problem aims to design a pricing scheme for a sequence of arriving buyers that maximizes the expected social welfare (or revenue) subject to different structural constraints. Inspired by applications with a hierarchy of service, this paper focuses on the cases where a laminar matroid characterizes the set of served buyers. We give the first Polynomial-Time Approximation Scheme (PTAS) for the problem when the laminar matroid has constant depth. Our approach is based on rounding the solution of a hierarchy of linear programming relaxations that approximate the optimum online solution with any degree of accuracy, plus a concentration argument showing that rounding incurs a small loss. We also study another variation, which we call the production-constrained problem. The allowable set of served buyers is characterized by a collection of production and shipping constraints that form a particular example of a laminar matroid. Using a similar LP-based approach, we design a PTAS for this problem, although in this special case the depth of the underlying laminar matroid is not necessarily a constant. The analysis exploits the negative dependency of the optimum selection rule in the lower levels of the laminar family. Finally, to demonstrate the generality of our technique, we employ the linear programming-based approach employed in the paper to re-derive some of the classic prophet inequalities known in the literature -- as a side result.

cs.GT↗

arXiv2024

Statistical Guarantees for Link Prediction using Graph Neural Networks

This paper derives statistical guarantees for the performance of Graph Neural Networks (GNNs) in link prediction tasks on graphs generated by a graphon. We propose a linear GNN architecture (LG-GNN) that produces consistent estimators for the underlying edge probabilities. We establish a bound on the mean squared error and give guarantees on the ability of LG-GNN to detect high-probability edges. Our guarantees hold for both sparse and dense graphs. Finally, we demonstrate some of the shortcomings of the classical GCN architecture, as well as verify our results on real and synthetic datasets.

cs.LG↗

arXiv2023

A Local Graph Limits Perspective on Sampling-Based GNNs

We propose a theoretical framework for training Graph Neural Networks (GNNs) on large input graphs via training on small, fixed-size sampled subgraphs. This framework is applicable to a wide range of models, including popular sampling-based GNNs, such as GraphSAGE and FastGCN. Leveraging the theory of graph local limits, we prove that, under mild assumptions, parameters learned from training sampling-based GNNs on small samples of a large input graph are within an $ε$-neighborhood of the outcome of training the same architecture on the whole graph. We derive bounds on the number of samples, the size of the graph, and the training steps required as a function of $ε$. Our results give a novel theoretical understanding for using sampling in training GNNs. They also suggest that by training GNNs on small samples of the input graph, practitioners can identify and select the best models, hyperparameters, and sampling algorithms more efficiently. We empirically illustrate our results on a node classification task on large citation graphs, observing that sampling-based GNNs trained on local subgraphs 12$\times$ smaller than the original graph achieve comparable performance to those trained on the input graph.

cs.LG↗

arXiv2022

Two-stage Stochastic Matching and Pricing with Applications to Ride Hailing

Matching and pricing are two critical levers in two-sided marketplaces to connect demand and supply. The platform can produce more efficient matching and pricing decisions by batching the demand requests. We initiate the study of the two-stage stochastic matching problem, with or without pricing, to enable the platform to make improved decisions in a batch with an eye toward the imminent future demand requests. This problem is motivated in part by applications in online marketplaces such as ride hailing platforms. We design online competitive algorithms for vertex-weighted (or unweighted) two-stage stochastic matching for maximizing supply efficiency, and two-stage joint matching and pricing for maximizing market efficiency. In the former problem, using a randomized primal-dual algorithm applied to a family of ``balancing'' convex programs, we obtain the optimal $3/4$ competitive ratio against the optimum offline benchmark. Using a factor revealing program and connections to submodular optimization, we improve this ratio against the optimum online benchmark to $(1-1/e+1/e^2)\approx 0.767$ for the unweighted and $0.761$ for the weighted case. In the latter problem, we design optimal $1/2$-competitive joint pricing and matching algorithm by borrowing ideas from the ex-ante prophet inequality literature. We also show an improved $(1-1/e)$-competitive algorithm for the special case of demand efficiency objective using the correlation gap of submodular functions. Finally, we complement our theoretical study by using DiDi's ride-sharing dataset for Chengdu city and numerically evaluating the performance of our proposed algorithms in practical instances of this problem.

cs.DS↗

arXiv2022

Sequential Submodular Maximization and Applications to Ranking an Assortment of Products

We study a submodular maximization problem motivated by applications in online retail. A platform displays a list of products to a user in response to a search query. The user inspects the first $k$ items in the list for a $k$ chosen at random from a given distribution, and decides whether to purchase an item from that set based on a choice model. The goal of the platform is to maximize the engagement of the shopper defined as the probability of purchase. This problem gives rise to a less-studied variation of submodular maximization in which we are asked to choose an $\textit{ordering}$ of a set of elements to maximize a linear combination of different submodular functions. First, using a reduction to maximizing submodular functions over matroids, we give an optimal $\left(1-1/e\right)$-approximation for this problem. We then consider a variant in which the platform cares not only about user engagement, but also about diversification across various groups of users, that is, guaranteeing a certain probability of purchase in each group. We characterize the polytope of feasible solutions and give a bi-criteria $((1-1/e)^2,(1-1/e)^2)$-approximation for this problem by rounding an approximate solution of a linear programming relaxation. For rounding, we rely on our reduction and the particular rounding techniques for matroid polytopes. For the special case in which underlying submodular functions are coverage functions -- which is practically relevant in online retail -- we propose an alternative LP relaxation and a simpler randomized rounding for the problem. This approach yields to an optimal bi-criteria $(1-1/e,1-1/e)$-approximation algorithm for the special case of the problem with coverage functions.

cs.GT↗

arXiv2022

Locality of Random Digraphs on Expanders

We study random digraphs on sequences of expanders with bounded average degree {which converge locally in probability}. We prove that the threshold for the existence of a giant strongly connected component, as well as the asymptotic fraction of nodes with giant fan-in or nodes with giant fan-out are local, in the sense that they are the same for two sequences with the same local limit. The digraph has a bow-tie structure, with all but a vanishing fraction of nodes lying either in the unique strongly connected giant and its fan-in and fan-out, or in sets with small fan-in and small fan-out. All local quantities are expressed in terms of percolation on the limiting rooted graph, without any structural assumptions on the limit, allowing, in particular, for non tree-like graphs. {In the course of establishing these results, we generalize previous results on the locality of the size of the giant to expanders of bounded average degree with possibly non-tree like limits. We also show that regardless of the local convergence of a sequence, the uniqueness of the giant and convergence of its relative size for unoriented percolation imply the bow-tie structure for directed percolation.} An application of our methods shows that the critical threshold for bond percolation and random digraphs on preferential attachment graphs is $p_c=0$, with an infinite order phase transition at $p_c$.

math.PR↗

arXiv2022

Sequential importance sampling for estimating expectations over the space of perfect matchings

This paper makes three contributions to estimating the number of perfect matching in bipartite graphs. First, we prove that the popular sequential importance sampling algorithm works in polynomial time for dense bipartite graphs. More carefully, our algorithm gives a $(1\pmε)$-approximation for the number of perfect matchings of a $λ$-dense bipartite graph, using $O(n^{\frac{1-2λ}λε^{-2}})$ samples. With size $n$ on each side and for $\frac{1}{2}>λ>0$, a $λ$-dense bipartite graph has all degrees greater than $(λ+\frac{1}{2})n$. Second, practical applications of the algorithm require many calls to matching algorithms. A novel preprocessing step is provided which makes significant improvements. Third, three applications are provided. The first is for counting Latin squares, the second is a practical way of computing the greedy algorithm for a card-guessing game with feedback, and the third is for stochastic block models. In all three examples, sequential importance sampling allows treating practical problems of reasonably large sizes.

math.PR↗

arXiv2022

Beating Greedy Matching in Sublinear Time

We study sublinear time algorithms for estimating the size of maximum matching in graphs. Our main result is a $(\frac{1}{2}+Ω(1))$-approximation algorithm which can be implemented in $O(n^{1+ε})$ time, where $n$ is the number of vertices and the constant $ε> 0$ can be made arbitrarily small. The best known lower bound for the problem is $Ω(n)$, which holds for any constant approximation. Existing algorithms either obtain the greedy bound of $\frac{1}{2}$-approximation [Behnezhad FOCS'21], or require some assumption on the maximum degree to run in $o(n^2)$-time [Yoshida, Yamamoto, and Ito STOC'09]. We improve over these by designing a less "adaptive" augmentation algorithm for maximum matching that might be of independent interest.

cs.DS↗