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Documents authored by Gajjar, Kshitij

Document
DOI: 10.4230/LIPIcs.FSTTCS.2023.11
Monotone Classes Beyond VNP

Authors: Prerona Chatterjee, Kshitij Gajjar, and Anamay Tengse

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

Abstract
In this work, we study the natural monotone analogues of various equivalent definitions of VPSPACE: a well studied class (Poizat 2008, Koiran & Perifel 2009, Malod 2011, Mahajan & Rao 2013) that is believed to be larger than VNP. We observe that these monotone analogues are not equivalent unlike their non-monotone counterparts, and propose monotone VPSPACE (mVPSPACE) to be defined as the monotone analogue of Poizat’s definition. With this definition, mVPSPACE turns out to be exponentially stronger than mVNP and also satisfies several desirable closure properties that the other analogues may not. Our initial goal was to understand the monotone complexity of transparent polynomials, a concept that was recently introduced by Hrubeš & Yehudayoff (2021). In that context, we show that transparent polynomials of large sparsity are hard for the monotone analogues of all the known definitions of VPSPACE, except for the one due to Poizat.
Cite as

Prerona Chatterjee, Kshitij Gajjar, and Anamay Tengse. Monotone Classes Beyond VNP. 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. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chatterjee_et_al:LIPIcs.FSTTCS.2023.11,
  author =	{Chatterjee, Prerona and Gajjar, Kshitij and Tengse, Anamay},
  title =	{{Monotone Classes Beyond VNP}},
  booktitle =	{43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)},
  pages =	{11:1--11:23},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.11},
  URN =		{urn:nbn:de:0030-drops-193846},
  doi =		{10.4230/LIPIcs.FSTTCS.2023.11},
  annote =	{Keywords: Algebraic Complexity, Monotone Computation, VPSPACE, Transparent Polynomials}
}
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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.ESA.2017.39
Distance-Preserving Subgraphs of Interval Graphs

Authors: Kshitij Gajjar and Jaikumar Radhakrishnan

Published in: LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Abstract
We consider the problem of finding small distance-preserving subgraphs of undirected, unweighted interval graphs that have k terminal vertices. We show that every interval graph admits a distance-preserving subgraph with O(k log k) branching vertices. We also prove a matching lower bound by exhibiting an interval graph based on bit-reversal permutation matrices. In addition, we show that interval graphs admit subgraphs with O(k) branching vertices that approximate distances up to an additive term of +1.
Cite as

Kshitij Gajjar and Jaikumar Radhakrishnan. Distance-Preserving Subgraphs of Interval Graphs. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 39:1-39:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{gajjar_et_al:LIPIcs.ESA.2017.39,
  author =	{Gajjar, Kshitij and Radhakrishnan, Jaikumar},
  title =	{{Distance-Preserving Subgraphs of Interval Graphs}},
  booktitle =	{25th Annual European Symposium on Algorithms (ESA 2017)},
  pages =	{39:1--39:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-049-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{87},
  editor =	{Pruhs, Kirk and Sohler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.39},
  URN =		{urn:nbn:de:0030-drops-78798},
  doi =		{10.4230/LIPIcs.ESA.2017.39},
  annote =	{Keywords: interval graphs, shortest path, distance-preserving subgraphs, bit-reversal permutation matrix}
}
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