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Documents authored by Gadekar, Ameet

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Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.6
Parameterized Approximation For Robust Clustering in Discrete Geometric Spaces

Authors: Fateme Abbasi, Sandip Banerjee, Jarosław Byrka, Parinya Chalermsook, Ameet Gadekar, Kamyar Khodamoradi, Dániel Marx, Roohani Sharma, and Joachim Spoerhase

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We consider the well-studied Robust (k,z)-Clustering problem, which generalizes the classic k-Median, k-Means, and k-Center problems and arises in the domains of robust optimization [Anthony, Goyal, Gupta, Nagarajan, Math. Oper. Res. 2010] and in algorithmic fairness [Abbasi, Bhaskara, Venkatasubramanian, 2021 & Ghadiri, Samadi, Vempala, 2022]. Given a constant z ≥ 1, the input to Robust (k,z)-Clustering is a set P of n points in a metric space (M,δ), a weight function w: P → ℝ_{≥ 0} and a positive integer k. Further, each point belongs to one (or more) of the m many different groups S_1,S_2,…,S_m ⊆ P. Our goal is to find a set X of k centers such that max_{i ∈ [m]} ∑_{p ∈ S_i} w(p) δ(p,X)^z is minimized. Complementing recent work on this problem, we give a comprehensive understanding of the parameterized approximability of the problem in geometric spaces where the parameter is the number k of centers. We prove the following results: [(i)] 1) For a universal constant η₀ > 0.0006, we devise a 3^z(1-η₀)-factor FPT approximation algorithm for Robust (k,z)-Clustering in discrete high-dimensional Euclidean spaces where the set of potential centers is finite. This shows that the lower bound of 3^z for general metrics [Goyal, Jaiswal, Inf. Proc. Letters, 2023] no longer holds when the metric has geometric structure. 2) We show that Robust (k,z)-Clustering in discrete Euclidean spaces is (√{3/2}- o(1))-hard to approximate for FPT algorithms, even if we consider the special case k-Center in logarithmic dimensions. This rules out a (1+ε)-approximation algorithm running in time f(k,ε)poly(m,n) (also called efficient parameterized approximation scheme or EPAS), giving a striking contrast with the recent EPAS for the continuous setting where centers can be placed anywhere in the space [Abbasi et al., FOCS'23]. 3) However, we obtain an EPAS for Robust (k,z)-Clustering in discrete Euclidean spaces when the dimension is sublogarithmic (for the discrete problem, earlier work [Abbasi et al., FOCS'23] provides an EPAS only in dimension o(log log n)). Our EPAS works also for metrics of sub-logarithmic doubling dimension.
Cite as

Fateme Abbasi, Sandip Banerjee, Jarosław Byrka, Parinya Chalermsook, Ameet Gadekar, Kamyar Khodamoradi, Dániel Marx, Roohani Sharma, and Joachim Spoerhase. Parameterized Approximation For Robust Clustering in Discrete Geometric Spaces. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{abbasi_et_al:LIPIcs.ICALP.2024.6,
  author =	{Abbasi, Fateme and Banerjee, Sandip and Byrka, Jaros{\l}aw and Chalermsook, Parinya and Gadekar, Ameet and Khodamoradi, Kamyar and Marx, D\'{a}niel and Sharma, Roohani and Spoerhase, Joachim},
  title =	{{Parameterized Approximation For Robust Clustering in Discrete Geometric Spaces}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{6:1--6:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.6},
  URN =		{urn:nbn:de:0030-drops-201494},
  doi =		{10.4230/LIPIcs.ICALP.2024.6},
  annote =	{Keywords: Clustering, approximation algorithms, parameterized complexity}
}
Document
DOI: 10.4230/LIPIcs.ESA.2016.11
On the Hardness of Learning Sparse Parities

Authors: Arnab Bhattacharyya, Ameet Gadekar, Suprovat Ghoshal, and Rishi Saket

Published in: LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Abstract
This work investigates the hardness of computing sparse solutions to systems of linear equations over F_2. Consider the k-EventSet problem: given a homogeneous system of linear equations over $\F_2$ on $n$ variables, decide if there exists a nonzero solution of Hamming weight at most k (i.e. a k-sparse solution). While there is a simple O(n^{k/2})-time algorithm for it, establishing fixed parameter intractability for k-EventSet has been a notorious open problem. Towards this goal, we show that unless \kclq can be solved in n^{o(k)} time, k-EventSet has no polynomial time algorithm when k = omega(log^2(n)). Our work also shows that the non-homogeneous generalization of the problem - which we call k-VectorSum - is W[1]-hard on instances where the number of equations is O(k*log(n)), improving on previous reductions which produced Omega(n) equations. We use the hardness of k-VectorSum as a starting point to prove the result for k-EventSet, and additionally strengthen the former to show the hardness of approximately learning k-juntas. In particular, we prove that given a system of O(exp(O(k))*log(n)) linear equations, it is W[1]-hard to decide if there is a k-sparse linear form satisfying all the equations or any function on at most k-variables (a k-junta) satisfies at most (1/2 + epsilon)-fraction of the equations, for any constant epsilon > 0. In the setting of computational learning, this shows hardness of approximate non-proper learning of k-parities. In a similar vein, we use the hardness of k-EventSet to show that that for any constant d, unless k-Clique can be solved in n^{o(k)} time, there is no poly(m,n)*2^{o(sqrt{k})} time algorithm to decide whether a given set of $m$ points in F_2^n satisfies: (i) there exists a non-trivial k-sparse homogeneous linear form evaluating to 0 on all the points, or (ii) any non-trivial degree d polynomial P supported on at most k variables evaluates to zero on approx Pr_{F_2^n}[P({z}) = 0] fraction of the points i.e., P is fooled by the set of points. Lastly, we study the approximation in the sparsity of the solution. Let the Gap-k-VectorSum problem be: given an instance of k-VectorSum of size n, decide if there exist a k-sparse solution, or every solution is of sparsity at least k' = (1+delta_0)k. Assuming the Exponential Time Hypothesis, we show that for some constants c_0, delta_0 > 0 there is no poly(n) time algorithm for Gap-k-VectorSum when k = omega((log(log( n)))^{c_0}).
Cite as

Arnab Bhattacharyya, Ameet Gadekar, Suprovat Ghoshal, and Rishi Saket. On the Hardness of Learning Sparse Parities. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bhattacharyya_et_al:LIPIcs.ESA.2016.11,
  author =	{Bhattacharyya, Arnab and Gadekar, Ameet and Ghoshal, Suprovat and Saket, Rishi},
  title =	{{On the Hardness of Learning Sparse Parities}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{11:1--11:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Sankowski, Piotr and Zaroliagis, Christos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.11},
  URN =		{urn:nbn:de:0030-drops-63628},
  doi =		{10.4230/LIPIcs.ESA.2016.11},
  annote =	{Keywords: Fixed Parameter Tractable, Juntas, Minimum Distance of Code, Psuedorandom Generators}
}
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