5 Search Results for "Jha, Agastya Vibhuti"

Document
DOI: 10.4230/LIPIcs.ESA.2025.111
Hardness of Median and Center in the Ulam Metric

Authors: Nick Fischer, Elazar Goldenberg, Mursalin Habib, and Karthik C. S.

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
The classical rank aggregation problem seeks to combine a set X of n permutations into a single representative "consensus" permutation. In this paper, we investigate two fundamental rank aggregation tasks under the well-studied Ulam metric: computing a median permutation (which minimizes the sum of Ulam distances to X) and computing a center permutation (which minimizes the maximum Ulam distance to X) in two settings. - Continuous Setting: In the continuous setting, the median/center is allowed to be any permutation. It is known that computing a center in the Ulam metric is NP-hard and we add to this by showing that computing a median is NP-hard as well via a simple reduction from the Max-Cut problem. While this result may not be unexpected, it had remained elusive until now and confirms a speculation by Chakraborty, Das, and Krauthgamer [SODA '21]. - Discrete Setting: In the discrete setting, the median/center must be a permutation from the input set. We fully resolve the fine-grained complexity of the discrete median and discrete center problems under the Ulam metric, proving that the naive Õ(n² L)-time algorithm (where L is the length of the permutation) is conditionally optimal. This resolves an open problem raised by Abboud, Bateni, Cohen-Addad, Karthik C. S., and Seddighin [APPROX '23]. Our reductions are inspired by the known fine-grained lower bounds for similarity measures, but we face and overcome several new highly technical challenges.
Cite as

Nick Fischer, Elazar Goldenberg, Mursalin Habib, and Karthik C. S.. Hardness of Median and Center in the Ulam Metric. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 111:1-111:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{fischer_et_al:LIPIcs.ESA.2025.111,
  author =	{Fischer, Nick and Goldenberg, Elazar and Habib, Mursalin and Karthik C. S.},
  title =	{{Hardness of Median and Center in the Ulam Metric}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{111:1--111:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.111},
  URN =		{urn:nbn:de:0030-drops-245809},
  doi =		{10.4230/LIPIcs.ESA.2025.111},
  annote =	{Keywords: Ulam distance, median, center, rank aggregation, fine-grained complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.50
New Results on a General Class of Minimum Norm Optimization Problems

Authors: Kuowen Chen, Jian Li, Yuval Rabani, and Yiran Zhang

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We study the general norm optimization for combinatorial problems, initiated by Chakrabarty and Swamy (STOC 2019). We propose a general formulation that captures a large class of combinatorial structures: we are given a set 𝒰 of n weighted elements and a family of feasible subsets ℱ. Each subset S ∈ ℱ is called a feasible solution/set of the problem. We denote the value vector by v = {v_i}_{i ∈ [n]}, where v_i ≥ 0 is the value of element i. For any subset S ⊆ 𝒰, we use v[S] to denote the n-dimensional vector {v_e⋅ 𝟏[e ∈ S]}_{e ∈ 𝒰} (i.e., we zero out all entries that are not in S). Let f: ℝⁿ → ℝ_+ be a symmetric monotone norm function. Our goal is to minimize the norm objective f(v[S]) over feasible subset S ∈ ℱ. The problem significantly generalizes the corresponding min-sum and min-max problems. We present a general equivalent reduction of the norm minimization problem to a multi-criteria optimization problem with logarithmic budget constraints, up to a constant approximation factor. Leveraging this reduction, we obtain constant factor approximation algorithms for the norm minimization versions of several covering problems, such as interval cover, multi-dimensional knapsack cover, and logarithmic factor approximation for set cover. We also study the norm minimization versions for perfect matching, s-t path and s-t cut. We show the natural linear programming relaxations for these problems have a large integrality gap. To complement the negative result, we show that, for perfect matching, it is possible to obtain a bi-criteria result: for any constant ε,δ > 0, we can find in polynomial time a nearly perfect matching (i.e., a matching that matches at least 1-ε proportion of vertices) and its cost is at most (8+δ) times of the optimum for perfect matching. Moreover, we establish the existence of a polynomial-time O(log log n)-approximation algorithm for the norm minimization variant of the s-t path problem. Specifically, our algorithm achieves an α-approximation with a time complexity of n^{O(log log n / α)}, where 9 ≤ α ≤ log log n.
Cite as

Kuowen Chen, Jian Li, Yuval Rabani, and Yiran Zhang. New Results on a General Class of Minimum Norm Optimization Problems. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 50:1-50:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chen_et_al:LIPIcs.ICALP.2025.50,
  author =	{Chen, Kuowen and Li, Jian and Rabani, Yuval and Zhang, Yiran},
  title =	{{New Results on a General Class of Minimum Norm Optimization Problems}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{50:1--50:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.50},
  URN =		{urn:nbn:de:0030-drops-234276},
  doi =		{10.4230/LIPIcs.ICALP.2025.50},
  annote =	{Keywords: Approximation Algorithms, Minimum Norm Optimization, Linear Programming}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.28
Coresets for 1-Center in 𝓁₁ Metrics

Authors: Amir Carmel, Chengzhi Guo, Shaofeng H.-C. Jiang, and Robert Krauthgamer

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
We explore the applicability of coresets - a small subset of the input dataset that approximates a predefined set of queries - to the 1-center problem in 𝓁₁ spaces. This approach could potentially extend to solving the 1-center problem in related metric spaces, and has implications for streaming and dynamic algorithms. We show that in 𝓁₁, unlike in Euclidean space, even weak coresets exhibit exponential dependency on the underlying dimension. Moreover, while inputs with a unique optimal center admit better bounds, they are not dimension independent. We then relax the guarantee of the coreset further, to merely approximate the value (optimal cost of 1-center), and obtain a dimension-independent coreset for every desired accuracy ε > 0. Finally, we discuss the broader implications of our findings to related metric spaces, and show explicit implications to Jaccard and Kendall’s tau distances.
Cite as

Amir Carmel, Chengzhi Guo, Shaofeng H.-C. Jiang, and Robert Krauthgamer. Coresets for 1-Center in 𝓁₁ Metrics. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{carmel_et_al:LIPIcs.ITCS.2025.28,
  author =	{Carmel, Amir and Guo, Chengzhi and Jiang, Shaofeng H.-C. and Krauthgamer, Robert},
  title =	{{Coresets for 1-Center in 𝓁₁ Metrics}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{28:1--28:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.28},
  URN =		{urn:nbn:de:0030-drops-226566},
  doi =		{10.4230/LIPIcs.ITCS.2025.28},
  annote =	{Keywords: clustering, k-center, minimum enclosing balls, coresets, 𝓁₁ norm, Kendall’s tau, Jaccard metric}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.91
A Sublinear Time Tester for Max-Cut on Clusterable Graphs

Authors: Agastya Vibhuti Jha and Akash Kumar

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

Abstract
One natural question in the area of sublinear time algorithms asks whether we can distinguish between graphs with max-cut value at least 1-ε from graphs with max-cut value at most 1/2+ε in the adjacency list model where we can make degree queries and neighbor queries. Chiplunkar, Kapralov, Khanna, Mousavifar, and Peres (FOCS' 18) showed that in graphs of bounded degree, one cannot hope for a factor 1/2+ε approximation to the max-cut value in time n^{1/2+o(ε)}. Recently, Peng and Yoshida (SODA '23) obtained o(n) time algorithms which can distinguish expanders with max-cut value at least 1-ε from expanders with small max-cut value (their running time is n^{1/2+O(ε)}). In this paper, going beyond the results of Peng-Yoshida, we develop sublinear time algorithms for this problem on clusterable graphs (which is a graph class with a good community structure). Our algorithms run in ≈ n^{0.5001+ O(ε)} time. A natural extension of Peng-Yoshida approach does not seem to work for clusterable graphs. Indeed, their random walk based technique tracks the 𝓁₂ length of random walk vectors and they exploit the difference in the length of these vectors to tell apart expanders with large cut value from expanders with small cut-value. Such approaches fail to be reliable when graph has loosely connected clusters. Taking inspiration from [Ashish Chiplunkar et al., 2018], we exploit the more refined geometry of spectra of clusterable graphs which leads to our sublinear time implementation. We prove a novel spectral lemma which shows that in a spectral expander 2 - λ_{n-1} ≥ Ω(λ₂). This lemma is leveraged to show that there is a suitable difference between spectra of clusterable graphs with large cut value and spectra of clusterable graphs with small cut value. We use this gap to obtain our sublinear time implementation. To do this, we obtain a nuanced understanding of the eigenvector structure of clusterable graphs and in particular, we show that the eigenvectors of the normalized Laplacian of a clusterable graph, corresponding to eigenvalues which are close to 2 have a small infinity norm.
Cite as

Agastya Vibhuti Jha and Akash Kumar. A Sublinear Time Tester for Max-Cut on Clusterable Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 91:1-91:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{jha_et_al:LIPIcs.ICALP.2024.91,
  author =	{Jha, Agastya Vibhuti and Kumar, Akash},
  title =	{{A Sublinear Time Tester for Max-Cut on Clusterable Graphs}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{91:1--91:17},
  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.2024.91},
  URN =		{urn:nbn:de:0030-drops-202344},
  doi =		{10.4230/LIPIcs.ICALP.2024.91},
  annote =	{Keywords: Sublinear Algorithms, Graph Algorithms, Clusterable Graphs, Property Testung}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2021.12
Approximating the Center Ranking Under Ulam

Authors: Diptarka Chakraborty, Kshitij Gajjar, and Agastya Vibhuti Jha

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

Abstract
We study the problem of approximating a center under the Ulam metric. The Ulam metric, defined over a set of permutations over [n], is the minimum number of move operations (deletion plus insertion) to transform one permutation into another. The Ulam metric is a simpler variant of the general edit distance metric. It provides a measure of dissimilarity over a set of rankings/permutations. In the center problem, given a set of permutations, we are asked to find a permutation (not necessarily from the input set) that minimizes the maximum distance to the input permutations. This problem is also referred to as maximum rank aggregation under Ulam. So far, we only know of a folklore 2-approximation algorithm for this NP-hard problem. Even for constantly many permutations, we do not know anything better than an exhaustive search over all n! permutations. In this paper, we achieve a (3/2 - 1/(3m))-approximation of the Ulam center in time n^O(m² ln m), for m input permutations over [n]. We therefore get a polynomial time bound while achieving better than a 3/2-approximation for constantly many permutations. This problem is of special interest even for constantly many permutations because under certain dissimilarity measures over rankings, even for four permutations, the problem is NP-hard. In proving our result, we establish a surprising connection between the approximate Ulam center problem and the closest string with wildcards problem (the center problem over the Hamming metric, allowing wildcards). We further study the closest string with wildcards problem and show that there cannot exist any (2-ε)-approximation algorithm (for any ε > 0) for it unless 𝖯 = NP. This inapproximability result is in sharp contrast with the same problem without wildcards, where we know of a PTAS.
Cite as

Diptarka Chakraborty, Kshitij Gajjar, and Agastya Vibhuti Jha. Approximating the Center Ranking Under Ulam. 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. 12:1-12:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chakraborty_et_al:LIPIcs.FSTTCS.2021.12,
  author =	{Chakraborty, Diptarka and Gajjar, Kshitij and Jha, Agastya Vibhuti},
  title =	{{Approximating the Center Ranking Under Ulam}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{12:1--12:21},
  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.2021.12},
  URN =		{urn:nbn:de:0030-drops-155230},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.12},
  annote =	{Keywords: Center Problem, Ulam Metric, Edit Distance, Closest String, Approximation Algorithms}
}
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