arXiv subjects

Debarati Das

Publications and source records attributed to Debarati Das.

At least 19 recordsLinked to original sources

arXiv2026

Ulam Rank Aggregation Is Hard to Approximate for Four Rankings

We study the approximability of rank aggregation under the Ulam metric. In the \emph{Ulam median} problem, the goal is to find a ranking (permutation) minimizing the sum of its Ulam distances to the input rankings, while in the \emph{Ulam center} problem, the objective is to minimize the maximum such distance. We prove that, for every $0<\varepsilon< 1/34$, it is $\mathrm{NP}$-hard to approximate either Ulam median or Ulam center within a factor of $35/34-\varepsilon$, even when the input consists of only four rankings. We further show that unless P = NP, neither problem admits a polynomial-time additive approximation scheme. Prior to our work, only the exact versions of both problems were known to be $\mathrm{NP}$-hard, and that too only when the number of input rankings is unbounded [Fischer et al., ESA'25 and Bachmaier et al., J. of Discrete Algorithms'15]. Furthermore, our inapproximability results are optimal in terms of the number of input rankings since for three inputs it is already known to be polynomial-time solvable [Chakraborty, Das, Krauthgamer, SODA'21]. En route, we introduce a new general framework for reducing Boolean constraint satisfaction problems (CSP) to the Ulam median with only four inputs. As a specific instantiation of the reduction framework, we obtain our hardness-of-approximation results. The corresponding hardness for the Ulam center follows from a reduction from the Ulam median.

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arXiv2026

Hardness of Approximation of Rank Aggregation on Ulam Metric

We study the approximability of rank aggregation under the Ulam metric. In the \emph{Ulam median} problem, the goal is to find a permutation minimizing the sum of its Ulam distances to the input permutations, while in the \emph{Ulam center} problem the objective is to minimize the maximum such distance. Both problems are known to be NP-hard, but no explicit approximation hardness was previously known. We prove that, for every $\varepsilon>0$, it is NP-hard to approximate either Ulam median or Ulam center within a factor of $51/50-\varepsilon$, even when the input consists of only four permutations. We further show that unless P = NP, neither problem admits a polynomial-time additive approximation scheme. The hardness result for Ulam median is established via a reduction from MAX-E3-LIN-2. The corresponding hardness for Ulam center is then obtained through a reduction from Ulam median.

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arXiv2026

A Near-Optimal Offline Algorithm for Dynamic All-Pairs Shortest Paths in Planar Digraphs

In the planar, dynamic All-Pairs Shortest Paths (APSP) problem, a planar, weighted digraph $G$ undergoes a sequence of edge weight updates and the goal is to maintain a data structure on $G$, that can quickly answer distance queries between any two vertices $x,y \in V(G)$. The currently best algorithms for this problem require $\tilde{O}(n^{2/3})$ worst-case update and query time, while conditional lower bounds show that either update or query time $n^{0.5-\delta}$ is needed for any constant $\delta > 0$. In this article, we present the first algorithm with near-optimal $\tilde{O}(\sqrt{n})$ worst-case update and query time for the offline setting, where the update sequence is given initially. This result is obtained by giving the first offline dynamic algorithm for maintaining dense distance graphs (DDGs) faster than recomputing from scratch after each update. Further, we also present an \emph{online} algorithm for the incremental APSP problem with $\tilde{O}(\sqrt{n})$ worst-case update/ query time. This allows us to reduce the online dynamic APSP problem to the online decremental APSP problem, which constitutes partial progress even for the online version of this notorious problem.

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arXiv2026

A Scalable and Unified Framework to Weighted Rank Aggregation

The rank aggregation problem seeks to combine multiple rank orderings of the same set of candidates into a single consensus ordering. Such problems arise in diverse domains, including web search, employment, college admissions, and voting. In this work we focus on the 1-median objective: given a set of m rankings over [n], the goal is to compute a ranking that minimizes the sum of its distances to all input rankings. We study rank aggregation under several classical distance metrics: Ulam distance, Spearman's footrule, Hamming distance, and Kendall-tau, as well as their weighted variants. Our contributions begin with a novel unified framework that identifies a key structural property: it suffices to focus on a small subset of rankings, where the corresponding local one-median provides a good approximation to the global median. This principle extends across these distance measures, yielding a general algorithmic framework for weighted rank aggregation. Building on this, we present a new approximation algorithm for rank aggregation under the Ulam distance that scales in the Massively Parallel Computation (MPC) model. Our algorithm computes a $(2-\alpha)$-approximation, for a constant $\alpha>0$, to the 1-median in a constant number of rounds, using local memory sublinear in n and total memory near-linear in n. We further design new MPC approximation algorithms for Spearman's footrule and for the element-weighted variants of Hamming and Kendall-tau distances. For each metric, we obtain a $(2-\zeta)$-approximation, for a constant $\zeta>0$, to the 1-median in a constant number of rounds, using local memory sublinear in n and total memory linear or near-linear in n. Moreover, for the Ulam distance, we simplify and strengthen the analysis of Chakraborty et al., obtaining an improved 1.968-approximation that further extends to the weighted setting.

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arXiv2026

A Generic Framework for Fair Consensus Clustering in Streams

Consensus clustering seeks to combine multiple clusterings of the same dataset, potentially derived by considering various non-sensitive attributes by different agents in a multi-agent environment, into a single partitioning that best reflects the overall structure of the underlying dataset. Recent work by Chakraborty et al, introduced a fair variant under proportionate fairness and obtained a constant-factor approximation by naively selecting the best closest fair input clustering; however, their offline approach requires storing all input clusterings, which is prohibitively expensive for most large-scale applications. In this paper, we initiate the study of fair consensus clustering in the streaming model, where input clusterings arrive sequentially and memory is limited. We design the first constant-factor algorithm that processes the stream while storing only a logarithmic number of inputs. En route, we introduce a new generic algorithmic framework that integrates closest fair clustering with cluster fitting, yielding improved approximation guarantees not only in the streaming setting but also when revisited offline. Furthermore, the framework is fairness-agnostic: it applies to any fairness definition for which an approximately close fair clustering can be computed efficiently. Finally, we extend our methods to the more general k-median consensus clustering problem.

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arXiv2025

Generalizing Fair Clustering to Multiple Groups: Algorithms and Applications

Clustering is a fundamental task in machine learning and data analysis, but it frequently fails to provide fair representation for various marginalized communities defined by multiple protected attributes -- a shortcoming often caused by biases in the training data. As a result, there is a growing need to enhance the fairness of clustering outcomes, ideally by making minimal modifications, possibly as a post-processing step after conventional clustering. Recently, Chakraborty et al. [COLT'25] initiated the study of \emph{closest fair clustering}, though in a restricted scenario where data points belong to only two groups. In practice, however, data points are typically characterized by many groups, reflecting diverse protected attributes such as age, ethnicity, gender, etc. In this work, we generalize the study of the \emph{closest fair clustering} problem to settings with an arbitrary number (more than two) of groups. We begin by showing that the problem is NP-hard even when all groups are of equal size -- a stark contrast with the two-group case, for which an exact algorithm exists. Next, we propose near-linear time approximation algorithms that efficiently handle arbitrary-sized multiple groups, thereby answering an open question posed by Chakraborty et al. [COLT'25]. Leveraging our closest fair clustering algorithms, we further achieve improved approximation guarantees for the \emph{fair correlation clustering} problem, advancing the state-of-the-art results established by Ahmadian et al. [AISTATS'20] and Ahmadi et al. [2020]. Additionally, we are the first to provide approximation algorithms for the \emph{fair consensus clustering} problem involving multiple (more than two) groups, thus addressing another open direction highlighted by Chakraborty et al. [COLT'25].

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arXiv2025

Dynamic Dyck and Tree Edit Distance: Decompositions and Reductions to String Edit Distance

We present the first dynamic algorithms for Dyck and tree edit distances with subpolynomial update times. Dyck edit distance measures how far a parenthesis string is from a well-parenthesized expression, while tree edit distance quantifies the minimum number of node insertions, deletions, and substitutions required to transform one rooted, ordered, labeled tree into another. Despite extensive study, no prior work has addressed efficient dynamic algorithms for these problems, which naturally arise in evolving structured data such as LaTeX documents, JSON or XML files, and RNA secondary structures. Our main contribution is a set of reductions and decompositions that transform Dyck and tree edit distance instances into efficiently maintainable string edit distance instances, which can be approximated within a $n^{o(1)}$ factor in $n^{o(1)}$ update time. For Dyck edit distance, our reduction incurs only polylogarithmic overheads in approximation and update time, yielding an $n^{o(1)}$-approximation with $n^{o(1)}$ updates. For tree edit distance, we introduce a new static reduction that improves the best-known approximation ratio from $n^{3/4}$ to $\tilde{O}(\sqrt{n})$ and removes the restriction to constant-degree trees. Extending this reduction dynamically achieves $n^{1/2+o(1)}$ approximation with $n^{o(1)}$ update time. A key component is a dynamic maintenance algorithm for history-independent heavy-light decompositions, of independent interest. We also provide a novel static and dynamic decomposition achieving an $O(k \log n)$-approximation when the tree edit distance is at most $k$. Combined with the trivial bound $k \le n$, this yields a dynamic deterministic $O(\sqrt{n \log n})$-approximation. In the static setting, our algorithm runs in near-linear time; dynamically, it requires only polylogarithmic updates, improving on prior linear-time static $O(\sqrt{n})$-approximation.

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arXiv2025

Towards Fair Representation: Clustering and Consensus

Consensus clustering, a fundamental task in machine learning and data analysis, aims to aggregate multiple input clusterings of a dataset, potentially based on different non-sensitive attributes, into a single clustering that best represents the collective structure of the data. In this work, we study this fundamental problem through the lens of fair clustering, as introduced by Chierichetti et al. [NeurIPS'17], which incorporates the disparate impact doctrine to ensure proportional representation of each protected group in the dataset within every cluster. Our objective is to find a consensus clustering that is not only representative but also fair with respect to specific protected attributes. To the best of our knowledge, we are the first to address this problem and provide a constant-factor approximation. As part of our investigation, we examine how to minimally modify an existing clustering to enforce fairness -- an essential postprocessing step in many clustering applications that require fair representation. We develop an optimal algorithm for datasets with equal group representation and near-linear time constant factor approximation algorithms for more general scenarios with different proportions of two group sizes. We complement our approximation result by showing that the problem is NP-hard for two unequal-sized groups. Given the fundamental nature of this problem, we believe our results on Closest Fair Clustering could have broader implications for other clustering problems, particularly those for which no prior approximation guarantees exist for their fair variants.

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arXiv2025

Prototypical Human-AI Collaboration Behaviors from LLM-Assisted Writing in the Wild

As large language models (LLMs) are used in complex writing workflows, users engage in multi-turn interactions to steer generations to better fit their needs. Rather than passively accepting output, users actively refine, explore, and co-construct text. We conduct a large-scale analysis of this collaborative behavior for users engaged in writing tasks in the wild with two popular AI assistants, Bing Copilot and WildChat. Our analysis goes beyond simple task classification or satisfaction estimation common in prior work and instead characterizes how users interact with LLMs through the course of a session. We identify prototypical behaviors in how users interact with LLMs in prompts following their original request. We refer to these as Prototypical Human-AI Collaboration Behaviors (PATHs) and find that a small group of PATHs explain a majority of the variation seen in user-LLM interaction. These PATHs span users revising intents, exploring texts, posing questions, adjusting style or injecting new content. Next, we find statistically significant correlations between specific writing intents and PATHs, revealing how users' intents shape their collaboration behaviors. We conclude by discussing the implications of our findings on LLM alignment.

cs.CL↗

arXiv2025

LawFlow: Collecting and Simulating Lawyers' Thought Processes on Business Formation Case Studies

Legal practitioners, particularly those early in their careers, face complex, high-stakes tasks that require adaptive, context-sensitive reasoning. While AI holds promise in supporting legal work, current datasets and models are narrowly focused on isolated subtasks and fail to capture the end-to-end decision-making required in real-world practice. To address this gap, we introduce LawFlow, a dataset of complete end-to-end legal workflows collected from trained law students, grounded in real-world business entity formation scenarios. Unlike prior datasets focused on input-output pairs or linear chains of thought, LawFlow captures dynamic, modular, and iterative reasoning processes that reflect the ambiguity, revision, and client-adaptive strategies of legal practice. Using LawFlow, we compare human and LLM-generated workflows, revealing systematic differences in structure, reasoning flexibility, and plan execution. Human workflows tend to be modular and adaptive, while LLM workflows are more sequential, exhaustive, and less sensitive to downstream implications. Our findings also suggest that legal professionals prefer AI to carry out supportive roles, such as brainstorming, identifying blind spots, and surfacing alternatives, rather than executing complex workflows end-to-end. Our results highlight both the current limitations of LLMs in supporting complex legal workflows and opportunities for developing more collaborative, reasoning-aware legal AI systems. All data and code are available on our project page (https://minnesotanlp.github.io/LawFlow-website/).

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arXiv2025

Fitting Tree Metrics and Ultrametrics in Data Streams

Fitting distances to tree metrics and ultrametrics are two widely used methods in hierarchical clustering, primarily explored within the context of numerical taxonomy. Given a positive distance function $D:\binom{V}{2}\rightarrow\mathbb{R}_{>0}$, the goal is to find a tree (or ultrametric) $T$ including all elements of set $V$ such that the difference between the distances among vertices in $T$ and those specified by $D$ is minimized. In this paper, we initiate the study of ultrametric and tree metric fitting problems in the semi-streaming model, where the distances between pairs of elements from $V$ (with $|V|=n$), defined by the function $D$, can arrive in an arbitrary order. We study these problems under various distance norms: For the $\ell_0$ objective, we provide a single-pass polynomial-time $\tilde{O}(n)$-space $O(1)$ approximation algorithm for ultrametrics and prove that no single-pass exact algorithm exists, even with exponential time. Next, we show that the algorithm for $\ell_0$ implies an $O(\Delta/\delta)$ approximation for the $\ell_1$ objective, where $\Delta$ is the maximum and $\delta$ is the minimum absolute difference between distances in the input. This bound matches the best-known approximation for the RAM model using a combinatorial algorithm when $\Delta/\delta=O(n)$. For the $\ell_\infty$ objective, we provide a complete characterization of the ultrametric fitting problem. We present a single-pass polynomial-time $\tilde{O}(n)$-space 2-approximation algorithm and show that no better than 2-approximation is possible, even with exponential time. We also show that, with an additional pass, it is possible to achieve a polynomial-time exact algorithm for ultrametrics. Finally, we extend the results for all these objectives to tree metrics by using only one additional pass through the stream and without asymptotically increasing the approximation factor.

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arXiv2024

Which Modality should I use -- Text, Motif, or Image? : Understanding Graphs with Large Language Models

Our research integrates graph data with Large Language Models (LLMs), which, despite their advancements in various fields using large text corpora, face limitations in encoding entire graphs due to context size constraints. This paper introduces a new approach to encoding a graph with diverse modalities, such as text, image, and motif, coupled with prompts to approximate a graph's global connectivity, thereby enhancing LLMs' efficiency in processing complex graph structures. The study also presents GraphTMI, a novel benchmark for evaluating LLMs in graph structure analysis, focusing on homophily, motif presence, and graph difficulty. Key findings indicate that the image modality, especially with vision-language models like GPT-4V, is superior to text in balancing token limits and preserving essential information and outperforms prior graph neural net (GNN) encoders. Furthermore, the research assesses how various factors affect the performance of each encoding modality and outlines the existing challenges and potential future developments for LLMs in graph understanding and reasoning tasks. All data will be publicly available upon acceptance.

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arXiv2024

Under the Surface: Tracking the Artifactuality of LLM-Generated Data

This work delves into the expanding role of large language models (LLMs) in generating artificial data. LLMs are increasingly employed to create a variety of outputs, including annotations, preferences, instruction prompts, simulated dialogues, and free text. As these forms of LLM-generated data often intersect in their application, they exert mutual influence on each other and raise significant concerns about the quality and diversity of the artificial data incorporated into training cycles, leading to an artificial data ecosystem. To the best of our knowledge, this is the first study to aggregate various types of LLM-generated text data, from more tightly constrained data like "task labels" to more lightly constrained "free-form text". We then stress test the quality and implications of LLM-generated artificial data, comparing it with human data across various existing benchmarks. Despite artificial data's capability to match human performance, this paper reveals significant hidden disparities, especially in complex tasks where LLMs often miss the nuanced understanding of intrinsic human-generated content. This study critically examines diverse LLM-generated data and emphasizes the need for ethical practices in data creation and when using LLMs. It highlights the LLMs' shortcomings in replicating human traits and behaviors, underscoring the importance of addressing biases and artifacts produced in LLM-generated content for future research and development. All data and code are available on our project page.

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arXiv2023

A Simple Algorithm for Multiple-Source Shortest Paths in Planar Digraphs

Given an $n$-vertex planar embedded digraph $G$ with non-negative edge weights and a face $f$ of $G$, Klein presented a data structure with $O(n\log n)$ space and preprocessing time which can answer any query $(u,v)$ for the shortest path distance in $G$ from $u$ to $v$ or from $v$ to $u$ in $O(\log n)$ time, provided $u$ is on $f$. This data structure is a key tool in a number of state-of-the-art algorithms and data structures for planar graphs. Klein's data structure relies on dynamic trees and the persistence technique as well as a highly non-trivial interaction between primal shortest path trees and their duals. The construction of our data structure follows a completely different and in our opinion very simple divide-and-conquer approach that solely relies on Single-Source Shortest Path computations and contractions in the primal graph. Our space and preprocessing time bound is $O(n\log |f|)$ and query time is $O(\log |f|)$ which is an improvement over Klein's data structure when $f$ has small size.

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arXiv2023

Balancing Effect of Training Dataset Distribution of Multiple Styles for Multi-Style Text Transfer

Text style transfer is an exciting task within the field of natural language generation that is often plagued by the need for high-quality paired datasets. Furthermore, training a model for multi-attribute text style transfer requires datasets with sufficient support across all combinations of the considered stylistic attributes, adding to the challenges of training a style transfer model. This paper explores the impact of training data input diversity on the quality of the generated text from the multi-style transfer model. We construct a pseudo-parallel dataset by devising heuristics to adjust the style distribution in the training samples. We balance our training dataset using marginal and joint distributions to train our style transfer models. We observe that a balanced dataset produces more effective control effects over multiple styles than an imbalanced or skewed one. Through quantitative analysis, we explore the impact of multiple style distributions in training data on style-transferred output. These findings will better inform the design of style-transfer datasets.

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arXiv2023

Weighted Edit Distance Computation: Strings, Trees and Dyck

Given two strings of length $n$ over alphabet $Σ$, and an upper bound $k$ on their edit distance, the algorithm of Myers (Algorithmica'86) and Landau and Vishkin (JCSS'88) computes the unweighted string edit distance in $\mathcal{O}(n+k^2)$ time. Till date, it remains the fastest algorithm for exact edit distance computation, and it is optimal under the Strong Exponential Hypothesis (STOC'15). Over the years, this result has inspired many developments, including fast approximation algorithms for string edit distance as well as similar $\tilde{\mathcal{O}}(n+$poly$(k))$-time algorithms for generalizations to tree and Dyck edit distances. Surprisingly, all these results hold only for unweighted instances. While unweighted edit distance is theoretically fundamental, almost all real-world applications require weighted edit distance, where different weights are assigned to different edit operations and may vary with the characters being edited. Given a weight function $w: Σ\cup \{\varepsilon \}\times Σ\cup \{\varepsilon \} \rightarrow \mathbb{R}_{\ge 0}$ (such that $w(a,a)=0$ and $w(a,b)\ge 1$ for all $a,b\in Σ\cup \{\varepsilon\}$ with $a\ne b$), the goal is to find an alignment that minimizes the total weight of edits. Except for the vanilla $\mathcal{O}(n^2)$-time dynamic-programming algorithm and its almost trivial $\mathcal{O}(nk)$-time implementation, none of the aforementioned developments on the unweighted edit distance apply to the weighted variant. In this paper, we propose the first $\mathcal{O}(n+$poly$(k))$-time algorithm that computes weighted string edit distance exactly, thus bridging a fundamental gap between our understanding of unweighted and weighted edit distance. We then generalize this result to weighted tree and Dyck edit distances, which lead to a deterministic algorithm that improves upon the previous work for unweighted tree edit distance.

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arXiv2022

Clustering Permutations: New Techniques with Streaming Applications

We study the classical metric $k$-median clustering problem over a set of input rankings (i.e., permutations), which has myriad applications, from social-choice theory to web search and databases. A folklore algorithm provides a $2$-approximate solution in polynomial time for all $k=O(1)$, and works irrespective of the underlying distance measure, so long it is a metric; however, going below the $2$-factor is a notorious challenge. We consider the Ulam distance, a variant of the well-known edit-distance metric, where strings are restricted to be permutations. For this metric, Chakraborty, Das, and Krauthgamer [SODA, 2021] provided a $(2-\delta)$-approximation algorithm for $k=1$, where $\delta\approx 2^{-40}$. Our primary contribution is a new algorithmic framework for clustering a set of permutations. Our first result is a $1.999$-approximation algorithm for the metric $k$-median problem under the Ulam metric, that runs in time $(k \log (nd))^{O(k)}n d^3$ for an input consisting of $n$ permutations over $[d]$. In fact, our framework is powerful enough to extend this result to the streaming model (where the $n$ input permutations arrive one by one) using only polylogarithmic (in $n$) space. Additionally, we show that similar results can be obtained even in the presence of outliers, which is presumably a more difficult problem.

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arXiv2022

$\tilde{O}(n+\mathrm{poly}(k))$-time Algorithm for Bounded Tree Edit Distance

Computing the edit distance of two strings is one of the most basic problems in computer science and combinatorial optimization. Tree edit distance is a natural generalization of edit distance in which the task is to compute a measure of dissimilarity between two (unweighted) rooted trees with node labels. Perhaps the most notable recent application of tree edit distance is in NoSQL big databases, such as MongoDB, where each row of the database is a JSON document represented as a labeled rooted tree, and finding dissimilarity between two rows is a basic operation. Until recently, the fastest algorithm for tree edit distance ran in cubic time (Demaine, Mozes, Rossman, Weimann; TALG'10); however, Mao (FOCS'21) broke the cubic barrier for the tree edit distance problem using fast matrix multiplication. Given a parameter $k$ as an upper bound on the distance, an $O(n+k^2)$-time algorithm for edit distance has been known since the 1980s due to the works of Myers (Algorithmica'86) and Landau and Vishkin (JCSS'88). The existence of an $\tilde{O}(n+\mathrm{poly}(k))$-time algorithm for tree edit distance has been posed as an open question, e.g., by Akmal and Jin (ICALP'21), who gave a state-of-the-art $\tilde{O}(nk^2)$-time algorithm. In this paper, we answer this question positively.

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