Stavros Sintos

Rahul Raychaudhury, Aryan Esmailpour, Sainyam Galhotra et al.

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.

Aryan Esmailpour, Stavros Sintos

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.

Rishi Advani, Abolfazl Asudeh, Mohsen Dehghankar et al.

The necklace splitting problem is a classic problem in fair division with many applications, including data-informed fair hash maps. We extend necklace splitting to a dynamic setting, allowing for relocation, insertion, and deletion of beads. We present linear-time, optimal algorithms for the two-color case that support all dynamic updates. For more than two colors, we give linear-time, optimal algorithms for relocation subject to a restriction on the number of agents. Finally, we propose a randomized algorithm for the two-color case that handles all dynamic updates, guarantees approximate fairness with high probability, and runs in polylogarithmic time when the number of agents is small.

Nima Shahbazi, Aryan Esmailpour, Stavros Sintos

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.