18 Search Results for "Inamdar, Tanmay"


Document
Exponential-Time Approximation Schemes via Compression

Authors: Tanmay Inamdar, Madhumita Kundu, Pekka Parviainen, M. S. Ramanujan, and Saket Saurabh

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)


Abstract
In this paper, we give a framework to design exponential-time approximation schemes for basic graph partitioning problems such as k-way cut, Multiway Cut, Steiner k-cut and Multicut, where the goal is to minimize the number of edges going across the parts. Our motivation to focus on approximation schemes for these problems comes from the fact that while it is possible to solve them exactly in 2^nn^{{𝒪}(1)} time (note that this is already faster than brute-forcing over all partitions or edge sets), it is not known whether one can do better. Using our framework, we design the first (1+ε)-approximation algorithms for the above problems that run in time 2^{f(ε)n} (for f(ε) < 1) for all these problems. As part of our framework, we present two compression procedures. The first of these is a "lossless" procedure, which is inspired by the seminal randomized contraction algorithm for Global Min-cut of Karger [SODA '93]. Here, we reduce the graph to an equivalent instance where the total number of edges is linearly bounded in the number of edges in an optimal solution of the original instance. Following this, we show how a careful combination of greedy choices and the best exact algorithm for the respective problems can exploit this structure and lead to our approximation schemes. Our first compression procedure bounds the number of edges linearly in the optimal solution, but this could still leave a dense graph as the solution size could be superlinear in the number of vertices. However, for several problems, it is known that they admit significantly faster algorithms on instances where solution size is linear in the number of vertices, in contrast to general instances. Hence, a natural question arises here. Could one reduce the solution size to linear in the number of vertices, at least in the case where we are willing to settle for a near-optimal solution, so that the aforementioned faster algorithms could be exploited? In the second compression procedure, using cut sparsifiers (this time, inspired by Benczúr and Karger [STOC '96]) we introduce "solution linearization" as a methodology to give an approximation-preserving reduction to the regime where solution size is linear in the number of vertices for certain cut problems. Using this, we obtain the first polynomial-space approximation schemes faster than 2^nn^{{𝒪}(1)} for Minimum bisection and Edge Bipartization. Along the way, we also design the first polynomial-space exact algorithms for these problems that run in time faster than 2^nn^{{𝒪}(1)}, in the regime where solution size is linear in the number of vertices. The use of randomized contraction and cut sparsifiers in the exponential-time setting is novel to the best of our knowledge and forms our conceptual contribution.

Cite as

Tanmay Inamdar, Madhumita Kundu, Pekka Parviainen, M. S. Ramanujan, and Saket Saurabh. Exponential-Time Approximation Schemes via Compression. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 64:1-64:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{inamdar_et_al:LIPIcs.ITCS.2024.64,
  author =	{Inamdar, Tanmay and Kundu, Madhumita and Parviainen, Pekka and Ramanujan, M. S. and Saurabh, Saket},
  title =	{{Exponential-Time Approximation Schemes via Compression}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{64:1--64:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.64},
  URN =		{urn:nbn:de:0030-drops-195920},
  doi =		{10.4230/LIPIcs.ITCS.2024.64},
  annote =	{Keywords: Exponential-Time Algorithms, Approximation Algorithms, Graph Algorithms, Cut Problems}
}
Document
FPT Approximations for Packing and Covering Problems Parameterized by Elimination Distance and Even Less

Authors: Tanmay Inamdar, Lawqueen Kanesh, Madhumita Kundu, M. S. Ramanujan, and Saket Saurabh

Published in: LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)


Abstract
For numerous graph problems in the realm of parameterized algorithms, using the size of a smallest deletion set (called a modulator) into well-understood graph families as parameterization has led to a long and successful line of research. Recently, however, there has been an extensive study of structural parameters that are potentially much smaller than the modulator size. In particular, recent papers [Jansen et al. STOC 2021; Agrawal et al. SODA 2022] have studied parameterization by the size of the modulator to a graph family ℋ(mod_ℋ(⋅)), elimination distance to ℋ(ed_ℋ(⋅)), and ℋ-treewidth (tw_ℋ(⋅)). These parameters are related by the fact that tw_ℋ lower bounds ed_ℋ, which in turn lower bounds mod_ℋ. While these new parameters have been successfully exploited to design fast exact algorithms their utility (especially that of ed_ℋ and tw_ℋ) in the context of approximation algorithms is mostly unexplored. The conceptual contribution of this paper is to present novel algorithmic meta-theorems that expand the impact of these structural parameters to the area of FPT Approximation, mirroring their utility in the design of exact FPT algorithms. Precisely, we show that if a covering or packing problem is definable in Monadic Second Order Logic and has a property called Finite Integer Index (FII), then the existence of an FPT Approximation Scheme (FPT-AS, i.e., (1±ε)-approximation) parameterized by mod_ℋ(⋅), ed_ℋ(⋅), and tw_ℋ(⋅) is in fact equivalent. As a consequence, we obtain FPT-ASes for a wide range of covering, packing, and domination problems on graphs with respect to these parameters. In the process, we show that several graph problems, that are W[1]-hard parameterized by mod_ℋ, admit FPT-ASes not only when parameterized by mod_ℋ, but even when parameterized by the potentially much smaller parameter tw_ℋ(⋅). In the spirit of [Agrawal et al. SODA 2022], our algorithmic results highlight a broader connection between these parameters in the world of approximation. As concrete exemplifications of our meta-theorems, we obtain FPT-ASes for well-studied graph problems such as Vertex Cover, Feedback Vertex Set, Cycle Packing and Dominating Set, parameterized by tw_ℋ(⋅) (and hence, also by mod_ℋ(⋅) or ed_ℋ(⋅)), where ℋ is any family of minor free graphs.

Cite as

Tanmay Inamdar, Lawqueen Kanesh, Madhumita Kundu, M. S. Ramanujan, and Saket Saurabh. FPT Approximations for Packing and Covering Problems Parameterized by Elimination Distance and Even Less. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{inamdar_et_al:LIPIcs.FSTTCS.2023.28,
  author =	{Inamdar, Tanmay and Kanesh, Lawqueen and Kundu, Madhumita and Ramanujan, M. S. and Saurabh, Saket},
  title =	{{FPT Approximations for Packing and Covering Problems Parameterized by Elimination Distance and Even Less}},
  booktitle =	{43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)},
  pages =	{28:1--28:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-304-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{284},
  editor =	{Bouyer, Patricia and Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.28},
  URN =		{urn:nbn:de:0030-drops-194013},
  doi =		{10.4230/LIPIcs.FSTTCS.2023.28},
  annote =	{Keywords: FPT-AS, F-Deletion, Packing, Elimination Distance, Elimination Treewidth}
}
Document
Kernelization for Spreading Points

Authors: Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Saket Saurabh, and Meirav Zehavi

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)


Abstract
We consider the following problem about dispersing points. Given a set of points in the plane, the task is to identify whether by moving a small number of points by small distance, we can obtain an arrangement of points such that no pair of points is "close" to each other. More precisely, for a family of n points, an integer k, and a real number d > 0, we ask whether at most k points could be relocated, each point at distance at most d from its original location, such that the distance between each pair of points is at least a fixed constant, say 1. A number of approximation algorithms for variants of this problem, under different names like distant representatives, disk dispersing, or point spreading, are known in the literature. However, to the best of our knowledge, the parameterized complexity of this problem remains widely unexplored. We make the first step in this direction by providing a kernelization algorithm that, in polynomial time, produces an equivalent instance with 𝒪(d²k³) points. As a byproduct of this result, we also design a non-trivial fixed-parameter tractable (FPT) algorithm for the problem, parameterized by k and d. Finally, we complement the result about polynomial kernelization by showing a lower bound that rules out the existence of a kernel whose size is polynomial in k alone, unless NP ⊆ coNP/poly.

Cite as

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Saket Saurabh, and Meirav Zehavi. Kernelization for Spreading Points. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 48:1-48:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{fomin_et_al:LIPIcs.ESA.2023.48,
  author =	{Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Saurabh, Saket and Zehavi, Meirav},
  title =	{{Kernelization for Spreading Points}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{48:1--48:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.48},
  URN =		{urn:nbn:de:0030-drops-187017},
  doi =		{10.4230/LIPIcs.ESA.2023.48},
  annote =	{Keywords: parameterized algorithms, kernelization, spreading points, distant representatives, unit disk packing}
}
Document
Parameterized Complexity of Fair Bisection: (FPT-Approximation meets Unbreakability)

Authors: Tanmay Inamdar, Daniel Lokshtanov, Saket Saurabh, and Vaishali Surianarayanan

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)


Abstract
In the Minimum Bisection problem input is a graph G and the goal is to partition the vertex set into two parts A and B, such that ||A|-|B|| ≤ 1 and the number k of edges between A and B is minimized. The problem is known to be NP-hard, and assuming the Unique Games Conjecture even NP-hard to approximate within a constant factor [Khot and Vishnoi, J.ACM'15]. On the other hand, a 𝒪(log n)-approximation algorithm [Räcke, STOC'08] and a parameterized algorithm [Cygan et al., ACM Transactions on Algorithms'20] running in time k^𝒪(k) n^𝒪(1) is known. The Minimum Bisection problem can be viewed as a clustering problem where edges represent similarity and the task is to partition the vertices into two equally sized clusters while minimizing the number of pairs of similar objects that end up in different clusters. Motivated by a number of egregious examples of unfair bias in AI systems, many fundamental clustering problems have been revisited and re-formulated to incorporate fairness constraints. In this paper we initiate the study of the Minimum Bisection problem with fairness constraints. Here the input is a graph G, positive integers c and k, a function χ:V(G) → {1, …, c} that assigns a color χ(v) to each vertex v in G, and c integers r_1,r_2,⋯,r_c. The goal is to partition the vertex set of G into two almost-equal sized parts A and B with at most k edges between them, such that for each color i ∈ {1, …, c}, A has exactly r_i vertices of color i. Each color class corresponds to a group which we require the partition (A, B) to treat fairly, and the constraints that A has exactly r_i vertices of color i can be used to encode that no group is over- or under-represented in either of the two clusters. We first show that introducing fairness constraints appears to make the Minimum Bisection problem qualitatively harder. Specifically we show that unless FPT=W[1] the problem admits no f(c)n^𝒪(1) time algorithm even when k = 0. On the other hand, our main technical contribution shows that is that this hardness result is simply a consequence of the very strict requirement that each color class i has exactly r_i vertices in A. In particular we give an f(k,c,ε)n^𝒪(1) time algorithm that finds a balanced partition (A, B) with at most k edges between them, such that for each color i ∈ [c], there are at most (1±ε)r_i vertices of color i in A. Our approximation algorithm is best viewed as a proof of concept that the technique introduced by [Lampis, ICALP'18] for obtaining FPT-approximation algorithms for problems of bounded tree-width or clique-width can be efficiently exploited even on graphs of unbounded width. The key insight is that the technique of Lampis is applicable on tree decompositions with unbreakable bags (as introduced in [Cygan et al., SIAM Journal on Computing'14]). An important ingredient of our approximation scheme is a combinatorial result that may be of independent interest, namely that for every k, every graph G admits a tree decomposition with adhesions of size at most 𝒪(k), unbreakable bags, and logarithmic depth.

Cite as

Tanmay Inamdar, Daniel Lokshtanov, Saket Saurabh, and Vaishali Surianarayanan. Parameterized Complexity of Fair Bisection: (FPT-Approximation meets Unbreakability). In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 63:1-63:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{inamdar_et_al:LIPIcs.ESA.2023.63,
  author =	{Inamdar, Tanmay and Lokshtanov, Daniel and Saurabh, Saket and Surianarayanan, Vaishali},
  title =	{{Parameterized Complexity of Fair Bisection: (FPT-Approximation meets Unbreakability)}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{63:1--63:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.63},
  URN =		{urn:nbn:de:0030-drops-187167},
  doi =		{10.4230/LIPIcs.ESA.2023.63},
  annote =	{Keywords: FPT Approximation, Minimum Bisection, Unbreakable Tree Decomposition, Treewidth}
}
Document
FPT Approximation and Subexponential Algorithms for Covering Few or Many Edges

Authors: Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, and Tomohiro Koana

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)


Abstract
We study the α-Fixed Cardinality Graph Partitioning (α-FCGP) problem, the generic local graph partitioning problem introduced by Bonnet et al. [Algorithmica 2015]. In this problem, we are given a graph G, two numbers k,p and 0 ≤ α ≤ 1, the question is whether there is a set S ⊆ V of size k with a specified coverage function cov_α(S) at least p (or at most p for the minimization version). The coverage function cov_α(⋅) counts edges with exactly one endpoint in S with weight α and edges with both endpoints in S with weight 1 - α. α-FCGP generalizes a number of fundamental graph problems such as Densest k-Subgraph, Max k-Vertex Cover, and Max (k,n-k)-Cut. A natural question in the study of α-FCGP is whether the algorithmic results known for its special cases, like Max k-Vertex Cover, could be extended to more general settings. One of the simple but powerful methods for obtaining parameterized approximation [Manurangsi, SOSA 2019] and subexponential algorithms [Fomin et al. IPL 2011] for Max k-Vertex Cover is based on the greedy vertex degree orderings. The main insight of our work is that the idea of greed vertex degree ordering could be used to design fixed-parameter approximation schemes (FPT-AS) for α > 0 and the subexponential-time algorithms for the problem on apex-minor free graphs for maximization with α > 1/3 and minimization with α < 1/3.

Cite as

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, and Tomohiro Koana. FPT Approximation and Subexponential Algorithms for Covering Few or Many Edges. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 46:1-46:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{fomin_et_al:LIPIcs.MFCS.2023.46,
  author =	{Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Koana, Tomohiro},
  title =	{{FPT Approximation and Subexponential Algorithms for Covering Few or Many Edges}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{46:1--46:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.46},
  URN =		{urn:nbn:de:0030-drops-185806},
  doi =		{10.4230/LIPIcs.MFCS.2023.46},
  annote =	{Keywords: Partial Vertex Cover, Approximation Algorithms, Max Cut}
}
Document
Fixed-Parameter Algorithms for Fair Hitting Set Problems

Authors: Tanmay Inamdar, Lawqueen Kanesh, Madhumita Kundu, Nidhi Purohit, and Saket Saurabh

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)


Abstract
Selection of a group of representatives satisfying certain fairness constraints, is a commonly occurring scenario. Motivated by this, we initiate a systematic algorithmic study of a fair version of Hitting Set. In the classical Hitting Set problem, the input is a universe 𝒰, a family ℱ of subsets of 𝒰, and a non-negative integer k. The goal is to determine whether there exists a subset S ⊆ 𝒰 of size k that hits (i.e., intersects) every set in ℱ. Inspired by several recent works, we formulate a fair version of this problem, as follows. The input additionally contains a family ℬ of subsets of 𝒰, where each subset in ℬ can be thought of as the group of elements of the same type. We want to find a set S ⊆ 𝒰 of size k that (i) hits all sets of ℱ, and (ii) does not contain too many elements of each type. We call this problem Fair Hitting Set, and chart out its tractability boundary from both classical as well as multivariate perspective. Our results use a multitude of techniques from parameterized complexity including classical to advanced tools, such as, methods of representative sets for matroids, FO model checking, and a generalization of best known kernels for Hitting Set.

Cite as

Tanmay Inamdar, Lawqueen Kanesh, Madhumita Kundu, Nidhi Purohit, and Saket Saurabh. Fixed-Parameter Algorithms for Fair Hitting Set Problems. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 55:1-55:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{inamdar_et_al:LIPIcs.MFCS.2023.55,
  author =	{Inamdar, Tanmay and Kanesh, Lawqueen and Kundu, Madhumita and Purohit, Nidhi and Saurabh, Saket},
  title =	{{Fixed-Parameter Algorithms for Fair Hitting Set Problems}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{55:1--55:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.55},
  URN =		{urn:nbn:de:0030-drops-185897},
  doi =		{10.4230/LIPIcs.MFCS.2023.55},
  annote =	{Keywords: Fairness, Parameterized Algorithms, Hitting Set}
}
Document
Coresets for Clustering in Geometric Intersection Graphs

Authors: Sayan Bandyapadhyay, Fedor V. Fomin, and Tanmay Inamdar

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)


Abstract
Designing coresets - small-space sketches of the data preserving cost of the solutions within (1± 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 for special metrics, like d-dimensional Euclidean space [Huang, Vishnoi, STOC'20], doubling metrics [Huang, Jiang, Li, Wu, FOCS'18], metrics of graphs of bounded treewidth [Baker, Braverman, Huang, Jiang, Krauthgamer, Wu, ICML’20], or graphs excluding a fixed minor [Braverman, Jiang, Krauthgamer, Wu, SODA’21]. In this paper, we provide the first constructions of coresets whose size does not depend on n for k-clustering in the metrics induced by geometric intersection graphs. For example, we obtain (k log²k)/ε^𝒪(1) size coresets for k-clustering in Euclidean-weighted unit-disk graphs (UDGs) and unit-square graphs (USGs). These constructions follow from a general theorem that identifies two canonical properties of a graph metric sufficient for obtaining coresets whose size is independent of n. 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 properties. Loosely speaking, the metrics of geometric intersection graphs are "similar" to the Euclidean metrics for points that are close, and to the shortest path metrics of planar graphs for points that are far apart. The main technical challenge in constructing centroid sets of small sizes is in combining these two very different metrics. 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).

Cite as

Sayan Bandyapadhyay, Fedor V. Fomin, and Tanmay Inamdar. Coresets for Clustering in Geometric Intersection Graphs. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{bandyapadhyay_et_al:LIPIcs.SoCG.2023.10,
  author =	{Bandyapadhyay, Sayan and Fomin, Fedor V. and Inamdar, Tanmay},
  title =	{{Coresets for Clustering in Geometric Intersection Graphs}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{10:1--10:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.10},
  URN =		{urn:nbn:de:0030-drops-178605},
  doi =		{10.4230/LIPIcs.SoCG.2023.10},
  annote =	{Keywords: k-median, k-means, clustering, coresets, geometric graphs}
}
Document
Exact Exponential Algorithms for Clustering Problems

Authors: Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Nidhi Purohit, and Saket Saurabh

Published in: LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)


Abstract
In this paper we initiate a systematic study of exact algorithms for some of the well known clustering problems, namely k-MEDIAN and k-MEANS. In k-MEDIAN, the input consists of a set X of n points belonging to a metric space, and the task is to select a subset C ⊆ X of k points as centers, such that the sum of the distances of every point to its nearest center is minimized. In k-MEANS, the objective is to minimize the sum of squares of the distances instead. It is easy to design an algorithm running in time max_{k ≤ n} {n choose k} n^𝒪(1) = 𝒪^*(2ⁿ) (here, 𝒪^*(⋅) notation hides polynomial factors in n). In this paper we design first non-trivial exact algorithms for these problems. In particular, we obtain an 𝒪^*((1.89)ⁿ) time exact algorithm for k-MEDIAN that works for any value of k. Our algorithm is quite general in that it does not use any properties of the underlying (metric) space - it does not even require the distances to satisfy the triangle inequality. In particular, the same algorithm also works for k-Means. We complement this result by showing that the running time of our algorithm is asymptotically optimal, up to the base of the exponent. That is, unless the Exponential Time Hypothesis fails, there is no algorithm for these problems running in time 2^o(n)⋅n^𝒪(1). Finally, we consider the "facility location" or "supplier" versions of these clustering problems, where, in addition to the set X we are additionally given a set of m candidate centers (or facilities) F, and objective is to find a subset of k centers from F. The goal is still to minimize the k-Median/k-Means/k-Center objective. For these versions we give a 𝒪(2ⁿ (mn)^𝒪(1)) time algorithms using subset convolution. We complement this result by showing that, under the Set Cover Conjecture, the "supplier" versions of these problems do not admit an exact algorithm running in time 2^{(1-ε) n} (mn)^𝒪(1).

Cite as

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Nidhi Purohit, and Saket Saurabh. Exact Exponential Algorithms for Clustering Problems. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{fomin_et_al:LIPIcs.IPEC.2022.13,
  author =	{Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Purohit, Nidhi and Saurabh, Saket},
  title =	{{Exact Exponential Algorithms for Clustering Problems}},
  booktitle =	{17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
  pages =	{13:1--13:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-260-0},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{249},
  editor =	{Dell, Holger and Nederlof, Jesper},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.13},
  URN =		{urn:nbn:de:0030-drops-173691},
  doi =		{10.4230/LIPIcs.IPEC.2022.13},
  annote =	{Keywords: clustering, k-median, k-means, exact algorithms}
}
Document
Track A: Algorithms, Complexity and Games
(Re)packing Equal Disks into Rectangle

Authors: Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, and Meirav Zehavi

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)


Abstract
The problem of packing of equal disks (or circles) into a rectangle is a fundamental geometric problem. (By a packing here we mean an arrangement of disks in a rectangle without overlapping.) We consider the following algorithmic generalization of the equal disk packing problem. In this problem, for a given packing of equal disks into a rectangle, the question is whether by changing positions of a small number of disks, we can allocate space for packing more disks. More formally, in the repacking problem, for a given set of n equal disks packed into a rectangle and integers k and h, we ask whether it is possible by changing positions of at most h disks to pack n+k disks. Thus the problem of packing equal disks is the special case of our problem with n = h = 0. While the computational complexity of packing equal disks into a rectangle remains open, we prove that the repacking problem is NP-hard already for h = 0. Our main algorithmic contribution is an algorithm that solves the repacking problem in time (h+k)^𝒪(h+k)⋅|I|^𝒪(1), where |I| is the input size. That is, the problem is fixed-parameter tractable parameterized by k and h.

Cite as

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, and Meirav Zehavi. (Re)packing Equal Disks into Rectangle. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 60:1-60:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{fomin_et_al:LIPIcs.ICALP.2022.60,
  author =	{Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Zehavi, Meirav},
  title =	{{(Re)packing Equal Disks into Rectangle}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{60:1--60:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.60},
  URN =		{urn:nbn:de:0030-drops-164011},
  doi =		{10.4230/LIPIcs.ICALP.2022.60},
  annote =	{Keywords: circle packing, unit disks, parameterized complexity, fixed-parameter tractability}
}
Document
Non-Uniform k-Center and Greedy Clustering

Authors: Tanmay Inamdar and Kasturi Varadarajan

Published in: LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)


Abstract
In the Non-Uniform k-Center (NUkC) problem, a generalization of the famous k-center clustering problem, we want to cover the given set of points in a metric space by finding a placement of balls with specified radii. In t-NUkC, we assume that the number of distinct radii is equal to t, and we are allowed to use k_i balls of radius r_i, for 1 ≤ i ≤ t. This problem was introduced by Chakrabarty et al. [ACM Trans. Alg. 16(4):46:1-46:19], who showed that a constant approximation for t-NUkC is not possible if t is unbounded, assuming 𝖯 ≠ NP. On the other hand, they gave a bicriteria approximation that violates the number of allowed balls as well as the given radii by a constant factor. They also conjectured that a constant approximation for t-NUkC should be possible if t is a fixed constant. Since then, there has been steady progress towards resolving this conjecture - currently, a constant approximation for 3-NUkC is known via the results of Chakrabarty and Negahbani [IPCO 2021], and Jia et al. [SOSA 2022]. We push the horizon by giving an O(1)-approximation for the Non-Uniform k-Center for 4 distinct types of radii. Our result is obtained via a novel combination of tools and techniques from the k-center literature, which also demonstrates that the different generalizations of k-center involving non-uniform radii, and multiple coverage constraints (i.e., colorful k-center), are closely interlinked with each other. We hope that our ideas will contribute towards a deeper understanding of the t-NUkC problem, eventually bringing us closer to the resolution of the CGK conjecture.

Cite as

Tanmay Inamdar and Kasturi Varadarajan. Non-Uniform k-Center and Greedy Clustering. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{inamdar_et_al:LIPIcs.SWAT.2022.28,
  author =	{Inamdar, Tanmay and Varadarajan, Kasturi},
  title =	{{Non-Uniform k-Center and Greedy Clustering}},
  booktitle =	{18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
  pages =	{28:1--28:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-236-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{227},
  editor =	{Czumaj, Artur and Xin, Qin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.28},
  URN =		{urn:nbn:de:0030-drops-161881},
  doi =		{10.4230/LIPIcs.SWAT.2022.28},
  annote =	{Keywords: k-center, approximation algorithms, non-uniform k-center, clustering}
}
Document
ETH Tight Algorithms for Geometric Intersection Graphs: Now in Polynomial Space

Authors: Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, and Saket Saurabh

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)


Abstract
De Berg et al. in [SICOMP 2020] gave an algorithmic framework for subexponential algorithms on geometric graphs with tight (up to ETH) running times. This framework is based on dynamic programming on graphs of weighted treewidth resulting in algorithms that use super-polynomial space. We introduce the notion of weighted treedepth and use it to refine the framework of de Berg et al. for obtaining polynomial space (with tight running times) on geometric graphs. As a result, we prove that for any fixed dimension d ≥ 2 on intersection graphs of similarly-sized fat objects many well-known graph problems including Independent Set, r-Dominating Set for constant r, Cycle Cover, Hamiltonian Cycle, Hamiltonian Path, Steiner Tree, Connected Vertex Cover, Feedback Vertex Set, and (Connected) Odd Cycle Transversal are solvable in time 2^𝒪(n^{1-1/d}) and within polynomial space.

Cite as

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, and Saket Saurabh. ETH Tight Algorithms for Geometric Intersection Graphs: Now in Polynomial Space. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{fomin_et_al:LIPIcs.FSTTCS.2021.21,
  author =	{Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Saurabh, Saket},
  title =	{{ETH Tight Algorithms for Geometric Intersection Graphs: Now in Polynomial Space}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{21:1--21:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.21},
  URN =		{urn:nbn:de:0030-drops-155323},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.21},
  annote =	{Keywords: Subexponential Algorithms, Geometric Intersection Graphs, Treedepth, Treewidth}
}
Document
Improved Bounds for Metric Capacitated Covering Problems

Authors: Sayan Bandyapadhyay

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)


Abstract
In the Metric Capacitated Covering (MCC) problem, given a set of balls ℬ in a metric space P with metric d and a capacity parameter U, the goal is to find a minimum sized subset ℬ' ⊆ ℬ and an assignment of the points in P to the balls in ℬ' such that each point is assigned to a ball that contains it and each ball is assigned with at most U points. MCC achieves an O(log |P|)-approximation using a greedy algorithm. On the other hand, it is hard to approximate within a factor of o(log |P|) even with β < 3 factor expansion of the balls. Bandyapadhyay et al. [SoCG 2018, DCG 2019] showed that one can obtain an O(1)-approximation for the problem with 6.47 factor expansion of the balls. An open question left by their work is to reduce the gap between the lower bound 3 and the upper bound 6.47. In this current work, we show that it is possible to obtain an O(1)-approximation with only 4.24 factor expansion of the balls. We also show a similar upper bound of 5 for a more generalized version of MCC for which the best previously known bound was 9.

Cite as

Sayan Bandyapadhyay. Improved Bounds for Metric Capacitated Covering Problems. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{bandyapadhyay:LIPIcs.ESA.2020.9,
  author =	{Bandyapadhyay, Sayan},
  title =	{{Improved Bounds for Metric Capacitated Covering Problems}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{9:1--9:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.9},
  URN =		{urn:nbn:de:0030-drops-128759},
  doi =		{10.4230/LIPIcs.ESA.2020.9},
  annote =	{Keywords: Capacitated covering, approximation algorithms, bicriteria approximation, LP rounding}
}
Document
Capacitated Sum-Of-Radii Clustering: An FPT Approximation

Authors: Tanmay Inamdar and Kasturi Varadarajan

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)


Abstract
In sum of radii clustering, the input consists of a finite set of points in a metric space. The problem asks to place a set of k balls centered at a subset of the points such that every point is covered by some ball, and the objective is to minimize the sum of radii of these balls. In the capacitated version of the problem, we want to assign each point to a ball containing it, such that no ball is assigned more than U points, where U denotes the capacity of the points. While constant approximations are known for the uncapacitated version of the problem, there is no work on the capacitated version. We make progress on this problem by obtaining a constant approximation using a Fixed Parameter Tractable (FPT) algorithm. In particular, the running time of the algorithm is of the form 2^O(k²) ⋅ n^O(1). As a warm-up for this result, we also give a constant approximation for the uncapacitated sum of radii clustering problem with matroid constraints, thus obtaining the first FPT approximation for this problem.

Cite as

Tanmay Inamdar and Kasturi Varadarajan. Capacitated Sum-Of-Radii Clustering: An FPT Approximation. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 62:1-62:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{inamdar_et_al:LIPIcs.ESA.2020.62,
  author =	{Inamdar, Tanmay and Varadarajan, Kasturi},
  title =	{{Capacitated Sum-Of-Radii Clustering: An FPT Approximation}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{62:1--62:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.62},
  URN =		{urn:nbn:de:0030-drops-129288},
  doi =		{10.4230/LIPIcs.ESA.2020.62},
  annote =	{Keywords: Sum-of-radii Clustering, Capacitated Clustering}
}
Document
APPROX
On Perturbation Resilience of Non-Uniform k-Center

Authors: Sayan Bandyapadhyay

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)


Abstract
The Non-Uniform k-center (NUkC) problem has recently been formulated by Chakrabarty, Goyal and Krishnaswamy [ICALP, 2016] as a generalization of the classical k-center clustering problem. In NUkC, given a set of n points P in a metric space and non-negative numbers r₁, r₂, … , r_k, the goal is to find the minimum dilation α and to choose k balls centered at the points of P with radius α⋅ r_i for 1 ≤ i ≤ k, such that all points of P are contained in the union of the chosen balls. They showed that the problem is NP-hard to approximate within any factor even in tree metrics. On the other hand, they designed a "bi-criteria" constant approximation algorithm that uses a constant times k balls. Surprisingly, no true approximation is known even in the special case when the r_i’s belong to a fixed set of size 3. In this paper, we study the NUkC problem under perturbation resilience, which was introduced by Bilu and Linial [Combinatorics, Probability and Computing, 2012]. We show that the problem under 2-perturbation resilience is polynomial time solvable when the r_i’s belong to a constant sized set. However, we show that perturbation resilience does not help in the general case. In particular, our findings imply that even with perturbation resilience one cannot hope to find any "good" approximation for the problem.

Cite as

Sayan Bandyapadhyay. On Perturbation Resilience of Non-Uniform k-Center. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 31:1-31:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{bandyapadhyay:LIPIcs.APPROX/RANDOM.2020.31,
  author =	{Bandyapadhyay, Sayan},
  title =	{{On Perturbation Resilience of Non-Uniform k-Center}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{31:1--31:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.31},
  URN =		{urn:nbn:de:0030-drops-126347},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.31},
  annote =	{Keywords: Non-Uniform k-center, stability, clustering, perturbation resilience}
}
Document
A Constant Approximation for Colorful k-Center

Authors: Sayan Bandyapadhyay, Tanmay Inamdar, Shreyas Pai, and Kasturi Varadarajan

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)


Abstract
In this paper, we consider the colorful k-center problem, which is a generalization of the well-known k-center problem. Here, we are given red and blue points in a metric space, and a coverage requirement for each color. The goal is to find the smallest radius rho, such that with k balls of radius rho, the desired number of points of each color can be covered. We obtain a constant approximation for this problem in the Euclidean plane. We obtain this result by combining a "pseudo-approximation" algorithm that works in any metric space, and an approximation algorithm that works for a special class of instances in the plane. The latter algorithm uses a novel connection to a certain matching problem in graphs.

Cite as

Sayan Bandyapadhyay, Tanmay Inamdar, Shreyas Pai, and Kasturi Varadarajan. A Constant Approximation for Colorful k-Center. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{bandyapadhyay_et_al:LIPIcs.ESA.2019.12,
  author =	{Bandyapadhyay, Sayan and Inamdar, Tanmay and Pai, Shreyas and Varadarajan, Kasturi},
  title =	{{A Constant Approximation for Colorful k-Center}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{12:1--12:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.12},
  URN =		{urn:nbn:de:0030-drops-111336},
  doi =		{10.4230/LIPIcs.ESA.2019.12},
  annote =	{Keywords: Colorful k-center, Euclidean Plane, LP Rounding, Outliers}
}
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