arXiv subjects

Aryan Esmailpour

Publications and source records attributed to Aryan Esmailpour.

10 recordsLinked to original sources

arXiv2026

Compact Representations of Geometric Bipartite Graphs via Weighted Biclique Covers

Bipartite graphs are a fundamental representation for relational data arising in recommendation systems, social networks, and communication graphs. A key challenge in these settings is to store and transmit large bipartite graphs compactly while preserving exact structural and path information. We study biclique-based representations of bipartite graphs $\boldsymbol{G}=(\boldsymbol{V},\boldsymbol{U},\boldsymbol{E})$, where the edge set is encoded using a collection of complete bipartite subgraphs. We focus on the Weighted Biclique Covering problem, which minimizes the total number of vertices used across all bicliques, and introduce a generalized variant that additionally penalizes the number of bicliques, capturing practical overheads in storage, transmission, and model complexity. While the weighted biclique covering problem is known to be $\mathsf{NP}$-Complete, we show that the generalized variant is also $\mathsf{NP}$-Complete. Despite this hardness, many real-world bipartite graphs admit low-dimensional geometric embeddings or can be well approximated by them. Leveraging this observation, we develop the first approximation algorithms with provable guarantees for the (generalized) weighted biclique covering problem on geometric bipartite graphs. Specifically, for $\delta$-disk bipartite graphs in low-dimensional $\ell_\infty^d$ spaces, we design a polynomial-time algorithm that achieves an $O(\log |\boldsymbol{U}| \cdot \log^d |\boldsymbol{V}|)$-approximation, combining ideas from greedy set cover, geometric range searching, and densest subgraph optimization. We also show how our algorithms extend to $\ell_\alpha^d$ metrics for any $\alpha\geq 1$. Finally, we evaluate our algorithms on real-world bipartite datasets and show that they efficiently compute significantly smaller biclique-based representations than natural baselines, while scaling to large graphs.

cs.DS↗

arXiv2026

Weighted Set Multi-Cover on Bounded Universe and Applications in Package Recommendation

The weighted set multi-cover problem is a fundamental generalization of set cover that arises in data-driven applications where one must select a small, low-cost subset from a large collection of candidates under coverage constraints. In data management settings, such problems arise naturally either as expressive database queries or as post-processing steps over query results, for example, when selecting representative or diverse subsets from large relations returned by database queries for decision support, recommendation, fairness-aware data selection, or crowd-sourcing. While the general weighted set multi-cover problem is NP-complete, many practical workloads involve a \emph{bounded universe} of items that must be covered, leading to the Weighted Set Multi-Cover with Bounded Universe (WSMC-BU) problem, where the universe size is constant. In this paper, we develop exact and approximation algorithms for WSMC-BU. We first discuss a dynamic programming algorithm that solves WSMC-BU exactly in $O(n^{\ell+1})$ time, where $n$ is the number of input sets and $\ell=O(1)$ is the universe size. We then present a $2$-approximation algorithm based on linear programming and rounding, running in $O(\mathcal{L}(n))$ time, where $\mathcal{L}(n)$ denotes the complexity of solving a linear program with $O(n)$ variables. To further improve efficiency for large datasets, we propose a faster $(2+\varepsilon)$-approximation algorithm with running time $O(n \log n + \mathcal{L}(\log W))$, where $W$ is the ratio of the total weight to the minimum weight, and $\varepsilon$ is an arbitrary constant specified by the user. Extensive experiments on real and synthetic datasets demonstrate that our methods consistently outperform greedy and standard LP-rounding baselines in both solution quality and runtime, making them suitable for data-intensive selection tasks over large query outputs.

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arXiv2026

Faster Relational Algorithms Using Geometric Data Structures

Optimization tasks over relational data, such as clustering, often suffer from the prohibitive cost of join operations, which are necessary to access the full dataset. While geometric data structures like BBD trees yield fast approximation algorithms in the standard computational setting, their application to relational data remains unclear due to the size of the join output. In this paper, we introduce a framework that leverages geometric insights to design faster algorithms when the data is stored as the results of a join query in a relational database. Our core contribution is the development of the RBBD tree, a randomized variant of the BBD tree tailored for relational settings. Instead of completely constructing the RBBD tree, by leveraging efficient sampling and counting techniques over relational joins, we enable on-the-fly efficient expansion of the RBBD tree, maintaining only the necessary parts. This allows us to simulate geometric query procedures without materializing the join result. As an application, we present algorithms that improve the state-of-the-art for relational $k$-center/means/median clustering by a factor of $k$ in running time while maintaining the same approximation guarantees. Our method is general and can be applied to various optimization problems in the relational setting.

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arXiv2026

Metric $k$-clustering using only Weak Comparison Oracles

Clustering is a fundamental primitive in unsupervised learning. However, classical algorithms for $k$-clustering (such as $k$-median and $k$-means) assume access to exact pairwise distances -- an unrealistic requirement in many modern applications. We study clustering in the \emph{Rank-model (R-model)}, where access to distances is entirely replaced by a \emph{quadruplet oracle} that provides only relative distance comparisons. In practice, such an oracle can represent learned models or human feedback, and is expected to be noisy and entail an access cost. Given a metric space with $n$ input items, we design randomized algorithms that, using only a noisy quadruplet oracle, compute a set of $O(k \cdot \mathsf{polylog}(n))$ centers along with a mapping from the input items to the centers such that the clustering cost of the mapping is at most constant times the optimum $k$-clustering cost. Our method achieves a query complexity of $O(n\cdot k \cdot \mathsf{polylog}(n))$ for arbitrary metric spaces and improves to $O((n+k^2) \cdot \mathsf{polylog}(n))$ when the underlying metric has bounded doubling dimension. When the metric has bounded doubling dimension we can further improve the approximation from constant to $1+\varepsilon$, for any arbitrarily small constant $\varepsilon\in(0,1)$, while preserving the same asymptotic query complexity. Our framework demonstrates how noisy, low-cost oracles, such as those derived from large language models, can be systematically integrated into scalable clustering algorithms.

cs.LG↗

arXiv2025

Subset Sampling over Joins

Subset sampling (also known as Poisson sampling), where the decision to include any specific element in the sample is made independently of all others, is a fundamental primitive in data analytics, enabling efficient approximation by processing representative subsets rather than massive datasets. While sampling from explicit lists is well-understood, modern applications -- such as machine learning over relational data -- often require sampling from a set defined implicitly by a relational join. In this paper, we study the problem of \emph{subset sampling over joins}: drawing a random subset from the join results, where each join result is included independently with some probability. We address the general setting where the probability is derived from input tuple weights via decomposable functions (e.g., product, sum, min, max). Since the join size can be exponentially larger than the input, the naive approach of materializing all join results to perform subset sampling is computationally infeasible. We propose the first efficient algorithms for subset sampling over acyclic joins: (1) a \emph{static index} for generating multiple (independent) subset samples over joins; (2) a \emph{one-shot} algorithm for generating a single subset sample over joins; (3) a \emph{dynamic index} that can support tuple insertions, while maintaining a one-shot sample or generating multiple (independent) samples. Our techniques achieve near-optimal time and space complexity with respect to the input size and the expected sample size.

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arXiv2025

Space-Time Tradeoffs for Spatial Conjunctive Queries

Given a conjunctive query and a database instance, we aim to develop an index that can efficiently answer spatial queries on the results of a conjunctive query. We are interested in some commonly used spatial queries, such as range emptiness, range count, and nearest neighbor queries. These queries have essential applications in data analytics, such as filtering relational data based on attribute ranges and temporal graph analysis for counting graph structures like stars, paths, and cliques. Furthermore, this line of research can accelerate relational algorithms that incorporate spatial queries in their workflow, such as relational clustering. Known approaches either have to spend $\tilde{O}(N)$ query time or use space as large as the number of query results, which are inefficient or unrealistic to employ in practice. Hence, we aim to construct an index that answers spatial conjunctive queries in both time- and space-efficient ways. In this paper, we establish lower bounds on the tradeoff between answering time and space usage. For $k$-star (resp. $k$-path) queries, we show that any index for range emptiness, range counting or nearest neighbor queries with $T$ answering time requires $\Omega\left(N+\frac{N^k}{T^k}\right)$ (resp. $\Omega\left(N+\frac{N^2}{T^{2/(k-1)}}\right)$) space. Then, we construct optimal indexes for answering range emptiness and range counting problems over $k$-star and $k$-path queries. Extending this result, we build an index for hierarchical queries. By resorting to the generalized hypertree decomposition, we can extend our index to arbitrary conjunctive queries for supporting spatial conjunctive queries. Finally, we show how our new indexes can be used to improve the running time of known algorithms in the relational setting.

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arXiv2025

A Theoretical Framework for Distribution-Aware Dataset Search

Effective data discovery is a cornerstone of modern data-driven decision-making. Yet, identifying datasets with specific distributional characteristics, such as percentiles or preferences, remains challenging. While recent proposals have enabled users to search based on percentile predicates, much of the research in data discovery relies on heuristics. This paper presents the first theoretically backed framework that unifies data discovery under centralized and decentralized settings. Let $\mathcal{P}=\{P_1,...,P_N\}$ be a repository of $N$ datasets, where $P_i\subset \mathbb{R}^d$, for $d=O(1)$ . We study the percentile indexing (Ptile) problem and the preference indexing (Pref) problem under the centralized and the federated setting. In the centralized setting we assume direct access to the datasets. In the federated setting we assume access to a synopsis of each dataset. The goal of Ptile is to construct a data structure such that given a predicate (rectangle $R$ and interval $θ$) report all indexes $J$ such that $j\in J$ iff $|P_j\cap R|/|P_j|\inθ$. The goal of Pref is to construct a data structure such that given a predicate (vector $v$ and interval $θ$) report all indexes $J$ such that $j\in J$ iff $ω(P_j,v)\in θ$, where $ω(P_j,v)$ is the inner-product of the $k$-th largest projection of $P_j$ on $v$. We first show that we cannot hope for near-linear data structures with polylogarithmic query time in the centralized setting. Next we show $\tilde{O}(N)$ space data structures that answer Ptile and Pref queries in $\tilde{O}(1+OUT)$ time, where $OUT$ is the output size. Each data structure returns a set of indexes $J$ such that i) for every $P_i$ that satisfies the predicate, $i\in J$ and ii) if $j\in J$ then $P_j$ satisfies the predicate up to an additive error $\varepsilon+2δ$, where $\varepsilon\in(0,1)$ and $δ$ is the error of synopses.

cs.DB↗

arXiv2024

Improved Approximation Algorithms for Relational Clustering

Clustering plays a crucial role in computer science, facilitating data analysis and problem-solving across numerous fields. By partitioning large datasets into meaningful groups, clustering reveals hidden structures and relationships within the data, aiding tasks such as unsupervised learning, classification, anomaly detection, and recommendation systems. Particularly in relational databases, where data is distributed across multiple tables, efficient clustering is essential yet challenging due to the computational complexity of joining tables. This paper addresses this challenge by introducing efficient algorithms for $k$-median and $k$-means clustering on relational data without the need for pre-computing the join query results. For the relational $k$-median clustering, we propose the first efficient relative approximation algorithm. For the relational $k$-means clustering, our algorithm significantly improves both the approximation factor and the running time of the known relational $k$-means clustering algorithms, which suffer either from large constant approximation factors, or expensive running time. Given a join query $Q$ and a database instance $D$ of $O(N)$ tuples, for both $k$-median and $k$-means clustering on the results of $Q$ on $D$, we propose randomized $(1+\varepsilon)\gamma$-approximation algorithms that run in roughly $O(k^2N^{\mathsf{fhw}})+T_\gamma(k^2)$ time, where $\varepsilon\in (0,1)$ is a constant parameter decided by the user, $\mathsf{fhw}$ is the fractional hyper-tree width of $Q$, while $\gamma$ and $T_\gamma(x)$ are respectively the approximation factor and the running time of a traditional clustering algorithm in the standard computational setting over $x$ points.

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arXiv2024

Combinatorial upper bounds for the smallest eigenvalue of a graph

Let $G$ be a graph, and let $λ(G)$ denote the smallest eigenvalue of $G$. First, we provide an upper bound for $λ(G)$ based on induced bipartite subgraphs of $G$. Consequently, we extract two other upper bounds, one relying on the average degrees of induced bipartite subgraphs and a more explicit one in terms of the chromatic number and the independence number of $G$. In particular, motivated by our bounds, we introduce two graph invariants that are of interest on their own. Finally, special attention goes to the investigation of the sharpness of our bounds in various classes of graphs as well as the comparison with an existing well-known upper bound.

math.CO↗

arXiv2023

Range (R\'enyi) Entropy Queries and Partitioning

Data partitioning that maximizes/minimizes the Shannon entropy, or more generally the R\'enyi entropy is a crucial subroutine in data compression, columnar storage, and cardinality estimation algorithms. These partition algorithms can be accelerated if we have a data structure to compute the entropy in different subsets of data when the algorithm needs to decide what block to construct. Such a data structure will also be useful for data analysts exploring different subsets of data to identify areas of interest. While it is generally known how to compute the Shannon or the R\'enyi entropy of a discrete distribution in the offline or streaming setting efficiently, we focus on the query setting where we aim to efficiently derive the entropy among a subset of data that satisfy some linear predicates. We solve this problem in a typical setting when we deal with real data, where data items are geometric points and each requested area is a query (hyper)rectangle. More specifically, we consider a set $P$ of $n$ weighted and colored points in $\mathbb{R}^d$, where $d$ is a constant. For the range S-entropy (resp. R-entropy) query problem, the goal is to construct a low space data structure, such that given a query (hyper)rectangle $R$, it computes the Shannon (resp. R\'enyi) entropy based on the colors and the weights of the points in $P\cap R$, in sublinear time. We show conditional lower bounds proving that we cannot hope for data structures with near-linear space and near-constant query time for both the range S-entropy and R-entropy query problems. Then, we propose exact data structures for $d=1$ and $d>1$ with $o(n^{2d})$ space and $o(n)$ query time for both problems. Finally, we propose near linear space data structures for returning either an additive or a multiplicative approximation of the Shannon (resp. R\'enyi) entropy in $P\cap R$.

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