arXiv subjects

Sayan Bandyapadhyay

Publications and source records attributed to Sayan Bandyapadhyay.

At least 19 recordsLinked to original sources

arXiv2025

Polynomial-Time Constant-Approximation for Fair Sum-of-Radii Clustering

In a seminal work, Chierichetti et al. introduced the $(t,k)$-fair clustering problem: Given a set of red points and a set of blue points in a metric space, a clustering is called fair if the number of red points in each cluster is at most $t$ times and at least $1/t$ times the number of blue points in that cluster. The goal is to compute a fair clustering with at most $k$ clusters that optimizes certain objective function. Considering this problem, they designed a polynomial-time $O(1)$- and $O(t)$-approximation for the $k$-center and the $k$-median objective, respectively. Recently, Carta et al. studied this problem with the sum-of-radii objective and obtained a $(6+\epsilon)$-approximation with running time $O((k\log_{1+\epsilon}(k/\epsilon))^kn^{O(1)})$, i.e., fixed-parameter tractable in $k$. Here $n$ is the input size. In this work, we design the first polynomial-time $O(1)$-approximation for $(t,k)$-fair clustering with the sum-of-radii objective, improving the result of Carta et al. Our result places sum-of-radii in the same group of objectives as $k$-center, that admit polynomial-time $O(1)$-approximations. This result also implies a polynomial-time $O(1)$-approximation for the Euclidean version of the problem, for which an $f(k)\cdot n^{O(1)}$-time $(1+\epsilon)$-approximation was known due to Drexler et al.. Here $f$ is an exponential function of $k$. We are also able to extend our result to any arbitrary $\ell\ge 2$ number of colors when $t=1$. This matches known results for the $k$-center and $k$-median objectives in this case. The significant disparity of sum-of-radii compared to $k$-center and $k$-median presents several complex challenges, all of which we successfully overcome in our work. Our main contribution is a novel cluster-merging-based analysis technique for sum-of-radii that helps us achieve the constant-approximation bounds.

cs.DS↗

arXiv2024

Robust Contraction Decomposition for Minor-Free Graphs and its Applications

We prove a robust contraction decomposition theorem for $H$-minor-free graphs, which states that given an $H$-minor-free graph $G$ and an integer $p$, one can partition in polynomial time the vertices of $G$ into $p$ sets $Z_1,\dots,Z_p$ such that $\operatorname{tw}(G/(Z_i \setminus Z')) = O(p + |Z'|)$ for all $i \in [p]$ and $Z' \subseteq Z_i$. Here, $\operatorname{tw}(\cdot)$ denotes the treewidth of a graph and $G/(Z_i \setminus Z')$ denotes the graph obtained from $G$ by contracting all edges with both endpoints in $Z_i \setminus Z'$. Our result generalizes earlier results by Klein [SICOMP 2008] and Demaine et al. [STOC 2011] based on partitioning $E(G)$, and some recent theorems for planar graphs by Marx et al. [SODA 2022], for bounded-genus graphs (more generally, almost-embeddable graphs) by Bandyapadhyay et al. [SODA 2022], and for unit-disk graphs by Bandyapadhyay et al. [SoCG 2022]. The robust contraction decomposition theorem directly results in parameterized algorithms with running time $2^{\widetilde{O}(\sqrt{k})} \cdot n^{O(1)}$ or $n^{O(\sqrt{k})}$ for every vertex/edge deletion problems on $H$-minor-free graphs that can be formulated as Permutation CSP Deletion or 2-Conn Permutation CSP Deletion. Consequently, we obtain the first subexponential-time parameterized algorithms for Subset Feedback Vertex Set, Subset Odd Cycle Transversal, Subset Group Feedback Vertex Set, 2-Conn Component Order Connectivity on $H$-minor-free graphs. For other problems which already have subexponential-time parameterized algorithms on $H$-minor-free graphs (e.g., Odd Cycle Transversal, Vertex Multiway Cut, Vertex Multicut, etc.), our theorem gives much simpler algorithms of the same running time.

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arXiv2024

Extraction Theorems With Small Extraction Numbers

In this work, we develop Extraction Theorems for classes of geometric objects with small extraction numbers. These classes include intervals, axis-parallel segments, axis-parallel rays, and octants. We investigate these classes of objects and prove small bounds on the extraction numbers. The tightness of these bounds is demonstrated by examples with matching lower bounds.

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arXiv2024

Fair Summarization: Bridging Quality and Diversity in Extractive Summaries

Fairness in multi-document summarization of user-generated content remains a critical challenge in natural language processing (NLP). Existing summarization methods often fail to ensure equitable representation across different social groups, leading to biased outputs. In this paper, we introduce two novel methods for fair extractive summarization: FairExtract, a clustering-based approach, and FairGPT, which leverages GPT-3.5-turbo with fairness constraints. We evaluate these methods using Divsumm summarization dataset of White-aligned, Hispanic, and African-American dialect tweets and compare them against relevant baselines. The results obtained using a comprehensive set of summarization quality metrics such as SUPERT, BLANC, SummaQA, BARTScore, and UniEval, as well as a fairness metric F, demonstrate that FairExtract and FairGPT achieve superior fairness while maintaining competitive summarization quality. Additionally, we introduce composite metrics (e.g., SUPERT+F, BLANC+F) that integrate quality and fairness into a single evaluation framework, offering a more nuanced understanding of the trade-offs between these objectives. Our code is available online.

cs.CL↗

arXiv2024

A Polynomial-Time Approximation for Pairwise Fair $k$-Median Clustering

In this work, we study pairwise fair clustering with $\ell \ge 2$ groups, where for every cluster $C$ and every group $i \in [\ell]$, the number of points in $C$ from group $i$ must be at most $t$ times the number of points in $C$ from any other group $j \in [\ell]$, for a given integer $t$. To the best of our knowledge, only bi-criteria approximation and exponential-time algorithms follow for this problem from the prior work on fair clustering problems when $\ell > 2$. In our work, focusing on the $\ell > 2$ case, we design the first polynomial-time $O(k^2\cdot \ell \cdot t)$-approximation for this problem with $k$-median cost that does not violate the fairness constraints. We complement our algorithmic result by providing hardness of approximation results, which show that our problem even when $\ell=2$ is almost as hard as the popular uniform capacitated $k$-median, for which no polynomial-time algorithm with an approximation factor of $o(\log k)$ is known.

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arXiv2024

An $O(n \log n)$-Time Approximation Scheme for Geometric Many-to-Many Matching

Geometric matching is an important topic in computational geometry and has been extensively studied over decades. In this paper, we study a geometric-matching problem, known as geometric many-to-many matching. In this problem, the input is a set $S$ of $n$ colored points in $\mathbb{R}^d$, which implicitly defines a graph $G = (S,E(S))$ where $E(S) = \{(p,q): p,q \in S \text{ have different colors}\}$, and the goal is to compute a minimum-cost subset $E^* \subseteq E(S)$ of edges that cover all points in $S$. Here the cost of $E^*$ is the sum of the costs of all edges in $E^*$, where the cost of a single edge $e$ is the Euclidean distance (or more generally, the $L_p$-distance) between the two endpoints of $e$. Our main result is a $(1+\varepsilon)$-approximation algorithm with an optimal running time $O_\varepsilon(n \log n)$ for geometric many-to-many matching in any fixed dimension, which works under any $L_p$-norm. This is the first near-linear approximation scheme for the problem in any $d \geq 2$. Prior to this work, only the bipartite case of geometric many-to-many matching was considered in $\mathbb{R}^1$ and $\mathbb{R}^2$, and the best known approximation scheme in $\mathbb{R}^2$ takes $O_\varepsilon(n^{1.5} \cdot \mathsf{poly}(\log n))$ time.

cs.CG↗

arXiv2024

FPT Constant-Approximations for Capacitated Clustering to Minimize the Sum of Cluster Radii

Clustering with capacity constraints is a fundamental problem that attracted significant attention throughout the years. In this paper, we give the first FPT constant-factor approximation algorithm for the problem of clustering points in a general metric into $k$ clusters to minimize the sum of cluster radii, subject to non-uniform hard capacity constraints. In particular, we give a $(15+ε)$-approximation algorithm that runs in $2^{0(k^2\log k)}\cdot n^3$ time. When capacities are uniform, we obtain the following improved approximation bounds: A (4 + $ε$)-approximation with running time $2^{O(k\log(k/ε))}n^3$, which significantly improves over the FPT 28-approximation of Inamdar and Varadarajan [ESA 2020]; a (2 + $ε$)-approximation with running time $2^{O(k/ε^2 \cdot\log(k/ε))}dn^3$ and a $(1+ε)$-approximation with running time $2^{O(kd\log ((k/ε)))}n^{3}$ in the Euclidean space; and a (1 + $ε$)-approximation in the Euclidean space with running time $2^{O(k/ε^2 \cdot\log(k/ε))}dn^3$ if we are allowed to violate the capacities by (1 + $ε$)-factor. We complement this result by showing that there is no (1 + $ε$)-approximation algorithm running in time $f(k)\cdot n^{O(1)}$, if any capacity violation is not allowed.

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arXiv2023

Euclidean Bottleneck Steiner Tree is Fixed-Parameter Tractable

In the Euclidean Bottleneck Steiner Tree problem, the input consists of a set of $n$ points in $\mathbb{R}^2$ called terminals and a parameter $k$, and the goal is to compute a Steiner tree that spans all the terminals and contains at most $k$ points of $\mathbb{R}^2$ as Steiner points such that the maximum edge-length of the Steiner tree is minimized, where the length of a tree edge is the Euclidean distance between its two endpoints. The problem is well-studied and is known to be NP-hard. In this paper, we give a $k^{O(k)} n^{O(1)}$-time algorithm for Euclidean Bottleneck Steiner Tree, which implies that the problem is fixed-parameter tractable (FPT). This settles an open question explicitly asked by Bae et al. [Algorithmica, 2011], who showed that the $\ell_1$ and $\ell_{\infty}$ variants of the problem are FPT. Our approach can be generalized to the problem with $\ell_p$ metric for any rational $1 \le p \le \infty$, or even other metrics on $\mathbb{R}^2$.

cs.CG↗

arXiv2023

On Colorful Vertex and Edge Cover Problems

In this paper, we study two generalizations of Vertex Cover and Edge Cover, namely Colorful Vertex Cover and Colorful Edge Cover. In the Colorful Vertex Cover problem, given an $n$-vertex edge-colored graph $G$ with colors from $\{1, \ldots, ω\}$ and coverage requirements $r_1, r_2, \ldots, r_ω$, the goal is to find a minimum-sized set of vertices that are incident on at least $r_i$ edges of color $i$, for each $1 \le i \le ω$, i.e., we need to cover at least $r_i$ edges of color $i$. Colorful Edge Cover is similar to Colorful Vertex Cover, except here we are given a vertex-colored graph and the goal is to cover at least $r_i$ vertices of color $i$, for each $1 \le i \le ω$, by a minimum-sized set of edges. These problems have several applications in fair covering and hitting of geometric set systems involving points and lines that are divided into multiple groups. Here, fairness ensures that the coverage (resp. hitting) requirement of every group is fully satisfied. We obtain a $(2+ε)$-approximation for the Colorful Vertex Cover problem in time $n^{O(ω/ε)}$. Thus, for a constant number of colors, the problem admits a $(2+ε)$-approximation in polynomial time. Next, for the Colorful Edge Cover problem, we design an $O(ωn^3)$ time exact algorithm, via a chain of reductions to a matching problem. For all intermediate problems in this chain of reductions, we design polynomial-time algorithms, which might be of independent interest.

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arXiv2023

Minimum-Membership Geometric Set Cover, Revisited

We revisit a natural variant of geometric set cover, called minimum-membership geometric set cover (MMGSC). In this problem, the input consists of a set $S$ of points and a set $\mathcal{R}$ of geometric objects, and the goal is to find a subset $\mathcal{R}^*\subseteq\mathcal{R}$ to cover all points in $S$ such that the \textit{membership} of $S$ with respect to $\mathcal{R}^*$, denoted by $\mathsf{memb}(S,\mathcal{R}^*)$, is minimized, where $\mathsf{memb}(S,\mathcal{R}^*)=\max_{p\in S}|\{R\in\mathcal{R}^*: p\in R\}|$. We achieve the following two main results. * We give the first polynomial-time constant-approximation algorithm for MMGSC with unit squares. This answers a question left open since the work of Erlebach and Leeuwen [SODA'08], who gave a constant-approximation algorithm with running time $n^{O(\mathsf{opt})}$ where $\mathsf{opt}$ is the optimum of the problem (i.e., the minimum membership). * We give the first polynomial-time approximation scheme (PTAS) for MMGSC with halfplanes. Prior to this work, it was even unknown whether the problem can be approximated with a factor of $o(\log n)$ in polynomial time, while it is well-known that the minimum-size set cover problem with halfplanes can be solved in polynomial time. We also consider a problem closely related to MMGSC, called minimum-ply geometric set cover (MPGSC), in which the goal is to find $\mathcal{R}^*\subseteq\mathcal{R}$ to cover $S$ such that the ply of $\mathcal{R}^*$ is minimized, where the ply is defined as the maximum number of objects in $\mathcal{R}^*$ which have a nonempty common intersection. Very recently, Durocher et al. gave the first constant-approximation algorithm for MPGSC with unit squares which runs in $O(n^{12})$ time. We give a significantly simpler constant-approximation algorithm with near-linear running time.

cs.CG↗

arXiv2023

Coresets for Clustering in Geometric Intersection Graphs

Designing coresets--small-space sketches of the data preserving cost of the solutions within $(1\pm ε)$-approximate factor--is an important research direction in the study of center-based $k$-clustering problems, such as $k$-means or $k$-median. Feldman and Langberg [STOC'11] have shown that for $k$-clustering of $n$ points in general metrics, it is possible to obtain coresets whose size depends logarithmically in $n$. Moreover, such a dependency in $n$ is inevitable in general metrics. A significant amount of recent work in the area is devoted to obtaining coresests whose sizes are independent of $n$ (i.e., ``small'' coresets) for special metrics, like $d$-dimensional Euclidean spaces, doubling metrics, metrics of graphs of bounded treewidth, or those excluding a fixed minor. In this paper, we provide the first constructions of small coresets for $k$-clustering in the metrics induced by geometric intersection graphs, such as Euclidean-weighted Unit Disk/Square Graphs. These constructions follow from a general theorem that identifies two canonical properties of a graph metric sufficient for obtaining small coresets. The proof of our theorem builds on the recent work of Cohen-Addad, Saulpic, and Schwiegelshohn [STOC '21], which ensures small-sized coresets conditioned on the existence of an interesting set of centers, called ``centroid set''. The main technical contribution of our work is the proof of the existence of such a small-sized centroid set for graphs that satisfy the two canonical geometric properties. The new coreset construction helps to design the first $(1+ε)$-approximation for center-based clustering problems in UDGs and USGs, that is fixed-parameter tractable in $k$ and $ε$ (FPT-AS).

cs.CG↗

arXiv2023

Proportionally Fair Matching with Multiple Groups

The study of fair algorithms has become mainstream in machine learning and artificial intelligence due to its increasing demand in dealing with biases and discrimination. Along this line, researchers have considered fair versions of traditional optimization problems including clustering, regression, ranking and voting. However, most of the efforts have been channeled into designing heuristic algorithms, which often do not provide any guarantees on the quality of the solution. In this work, we study matching problems with the notion of proportional fairness. Proportional fairness is one of the most popular notions of group fairness where every group is represented up to an extent proportional to the final selection size. Matching with proportional fairness or more commonly, proportionally fair matching, was introduced in [Chierichetti et al., AISTATS, 2019], where the problem was studied with only two groups. However, in many practical applications, the number of groups -- although often a small constant -- is larger than two. In this work, we make the first step towards understanding the computational complexity of proportionally fair matching with more than two groups. We design exact and approximation algorithms achieving reasonable guarantees on the quality of the matching as well as on the time complexity. Our algorithms are also supported by suitable hardness bounds.

cs.DS↗

arXiv2022

Geometric Planar Networks on Bichromatic Points

We study three classical graph problems - Hamiltonian path, minimum spanning tree, and minimum perfect matching on geometric graphs induced by bichromatic (red and blue) points. These problems have been widely studied for points in the Euclidean plane, and many of them are NP-hard. In this work, we consider these problems for collinear points. We show that almost all of these problems can be solved in linear time in this setting.

cs.CG↗

arXiv2022

Subexponential Parameterized Algorithms for Cut and Cycle Hitting Problems on H-Minor-Free Graphs

We design the first subexponential-time (parameterized) algorithms for several cut and cycle-hitting problems on $H$-minor free graphs. In particular, we obtain the following results (where $k$ is the solution-size parameter). 1. $2^{O(\sqrt{k}\log k)} \cdot n^{O(1)}$ time algorithms for Edge Bipartization and Odd Cycle Transversal; 2. a $2^{O(\sqrt{k}\log^4 k)} \cdot n^{O(1)}$ time algorithm for Edge Multiway Cut and a $2^{O(r \sqrt{k} \log k)} \cdot n^{O(1)}$ time algorithm for Vertex Multiway Cut, where $r$ is the number of terminals to be separated; 3. a $2^{O((r+\sqrt{k})\log^4 (rk))} \cdot n^{O(1)}$ time algorithm for Edge Multicut and a $2^{O((\sqrt{rk}+r) \log (rk))} \cdot n^{O(1)}$ time algorithm for Vertex Multicut, where $r$ is the number of terminal pairs to be separated; 4. a $2^{O(\sqrt{k} \log g \log^4 k)} \cdot n^{O(1)}$ time algorithm for Group Feedback Edge Set and a $2^{O(g \sqrt{k}\log(gk))} \cdot n^{O(1)}$ time algorithm for Group Feedback Vertex Set, where $g$ is the size of the group. 5. In addition, our approach also gives $n^{O(\sqrt{k})}$ time algorithms for all above problems with the exception of $n^{O(r+\sqrt{k})}$ time for Edge/Vertex Multicut and $(ng)^{O(\sqrt{k})}$ time for Group Feedback Edge/Vertex Set. We obtain our results by giving a new decomposition theorem on graphs of bounded genus, or more generally, an $h$-almost-embeddable graph for any fixed constant $h$. In particular we show the following. Let $G$ be an $h$-almost-embeddable graph for a constant $h$. Then for every $p\in\mathbb{N}$, there exist disjoint sets $Z_1,\dots,Z_p \subseteq V(G)$ such that for every $i \in \{1,\dots,p\}$ and every $Z'\subseteq Z_i$, the treewidth of $G/(Z_i\backslash Z')$ is $O(p+|Z'|)$. Here $G/(Z_i\backslash Z')$ is the graph obtained from $G$ by contracting edges with both endpoints in $Z_i \backslash Z'$.

cs.DS↗

arXiv2021

Parameterized Approximation Algorithms for $k$-Center Clustering and Variants

$k$-center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.93, even in the plane, if one insists the dependence on $k$ in the running time be polynomial. Without this restriction, a classic algorithm yields a $2^{O((k\log k)/ε)}dn$-time $(1+ε)$-approximation for Euclidean $k$-center, where $d$ is the dimension. We give a faster algorithm for small dimensions: roughly speaking an $O^*(2^{O((1/ε)^{O(d)} \cdot k^{1-1/d} \cdot \log k)})$-time $(1+ε)$-approximation. In particular, the running time is roughly $O^*(2^{O((1/ε)^{O(1)}\sqrt{k}\log k)})$ in the plane. We complement our algorithmic result with a matching hardness lower bound. We also consider a well-studied generalization of $k$-center, called Non-uniform $k$-center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a $2^{O(k\log k)}n^2$ time $3$-approximation for NUkC in general metrics, and a $2^{O((k\log k)/ε)}dn$ time $(1+ε)$-approximation for Euclidean NUkC. The latter time bound matches the bound for $k$-center.

cs.DS↗

arXiv2021

How to Find a Good Explanation for Clustering?

$k$-means and $k$-median clustering are powerful unsupervised machine learning techniques. However, due to complicated dependences on all the features, it is challenging to interpret the resulting cluster assignments. Moshkovitz, Dasgupta, Rashtchian, and Frost [ICML 2020] proposed an elegant model of explainable $k$-means and $k$-median clustering. In this model, a decision tree with $k$ leaves provides a straightforward characterization of the data set into clusters. We study two natural algorithmic questions about explainable clustering. (1) For a given clustering, how to find the "best explanation" by using a decision tree with $k$ leaves? (2) For a given set of points, how to find a decision tree with $k$ leaves minimizing the $k$-means/median objective of the resulting explainable clustering? To address the first question, we introduce a new model of explainable clustering. Our model, inspired by the notion of outliers in robust statistics, is the following. We are seeking a small number of points (outliers) whose removal makes the existing clustering well-explainable. For addressing the second question, we initiate the study of the model of Moshkovitz et al. from the perspective of multivariate complexity. Our rigorous algorithmic analysis sheds some light on the influence of parameters like the input size, dimension of the data, the number of outliers, the number of clusters, and the approximation ratio, on the computational complexity of explainable clustering.

cs.DS↗

arXiv2021

Exact and Approximation Algorithms for Many-To-Many Point Matching in the Plane

Given two sets $S$ and $T$ of points in the plane, of total size $n$, a {many-to-many} matching between $S$ and $T$ is a set of pairs $(p,q)$ such that $p\in S$, $q\in T$ and for each $r\in S\cup T$, $r$ appears in at least one such pair. The {cost of a pair} $(p,q)$ is the (Euclidean) distance between $p$ and $q$. In the {minimum-cost many-to-many matching} problem, the goal is to compute a many-to-many matching such that the sum of the costs of the pairs is minimized. This problem is a restricted version of minimum-weight edge cover in a bipartite graph, and hence can be solved in $O(n^3)$ time. In a more restricted setting where all the points are on a line, the problem can be solved in $O(n\log n)$ time [Colannino, Damian, Hurtado, Langerman, Meijer, Ramaswami, Souvaine, Toussaint; Graphs Comb., 2007]. However, no progress has been made in the general planar case in improving the cubic time bound. In this paper, we obtain an $O(n^2\cdot poly(\log n))$ time exact algorithm and an $O( n^{3/2}\cdot poly(\log n))$ time $(1+ε)$-approximation in the planar case. Our results affirmatively address an open problem posed in [Colannino et al., Graphs Comb., 2007].

cs.CG↗

arXiv2021

Parameterized Complexity of Feature Selection for Categorical Data Clustering

We develop new algorithmic methods with provable guarantees for feature selection in regard to categorical data clustering. While feature selection is one of the most common approaches to reduce dimensionality in practice, most of the known feature selection methods are heuristics. We study the following mathematical model. We assume that there are some inadvertent (or undesirable) features of the input data that unnecessarily increase the cost of clustering. Consequently, we want to select a subset of the original features from the data such that there is a small-cost clustering on the selected features. More precisely, for given integers $\ell$ (the number of irrelevant features) and $k$ (the number of clusters), budget $B$, and a set of $n$ categorical data points (represented by $m$-dimensional vectors whose elements belong to a finite set of values $Σ$), we want to select $m-\ell$ relevant features such that the cost of any optimal $k$-clustering on these features does not exceed $B$. Here the cost of a cluster is the sum of Hamming distances ($\ell_0$-distances) between the selected features of the elements of the cluster and its center. The clustering cost is the total sum of the costs of the clusters. We use the framework of parameterized complexity to identify how the complexity of the problem depends on parameters $k$, $B$, and $|Σ|$. Our main result is an algorithm that solves the Feature Selection problem in time $f(k,B,|Σ|)\cdot m^{g(k,|Σ|)}\cdot n^2$ for some functions $f$ and $g$. In other words, the problem is fixed-parameter tractable parameterized by $B$ when $|Σ|$ and $k$ are constants. Our algorithm is based on a solution to a more general problem, Constrained Clustering with Outliers. We also complement our algorithmic findings with complexity lower bounds.

cs.DS↗