3 Search Results for "Kumar, Rajendra"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.107
Improved Lower Bounds for Approximating Parameterized Nearest Codeword and Related Problems Under ETH

Authors: Shuangle Li, Bingkai Lin, and Yuwei Liu

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

Abstract
In this paper we present a new gap-creating randomized self-reduction for the parameterized Maximum Likelihood Decoding problem over 𝔽_p (k-MLD_p). The reduction takes a k-MLD_p instance with k⋅ n d-dimensional vectors as input, runs in O(d2^{O(k)}n^{1.01}) time for some computable function f, outputs a (3/2-ε)-Gap-k'-MLD_p instance for any ε > 0, where k' = O(k²log k). Using this reduction, we show that assuming the randomized Exponential Time Hypothesis (ETH), no algorithms can approximate k-MLD_p (and therefore its dual problem k-NCP_p) within factor (3/2-ε) in f(k)⋅ n^{o(√{k/log k})} time for any ε > 0. We then use reduction by Bhattacharyya, Ghoshal, Karthik and Manurangsi (ICALP 2018) to amplify the (3/2-ε)-gap to any constant. As a result, we show that assuming ETH, no algorithms can approximate k-NCP_p and k-MDP_p within γ-factor in f(k)⋅ n^{o(k^{ε_γ})} time for some constant ε_γ > 0. Combining with the gap-preserving reduction by Bennett, Cheraghchi, Guruswami and Ribeiro (STOC 2023), we also obtain similar lower bounds for k-MDP_p, k-CVP_p and k-SVP_p. These results improve upon the previous f(k)⋅ n^{Ω(poly log k)} lower bounds for these problems under ETH using reductions by Bhattacharyya et al. (J.ACM 2021) and Bennett et al. (STOC 2023).
Cite as

Shuangle Li, Bingkai Lin, and Yuwei Liu. Improved Lower Bounds for Approximating Parameterized Nearest Codeword and Related Problems Under ETH. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 107:1-107:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{li_et_al:LIPIcs.ICALP.2024.107,
  author =	{Li, Shuangle and Lin, Bingkai and Liu, Yuwei},
  title =	{{Improved Lower Bounds for Approximating Parameterized Nearest Codeword and Related Problems Under ETH}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{107:1--107:20},
  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.2024.107},
  URN =		{urn:nbn:de:0030-drops-202500},
  doi =		{10.4230/LIPIcs.ICALP.2024.107},
  annote =	{Keywords: Nearest Codeword Problem, Hardness of Approximations, Fine-grained Complexity, Parameterized Complexity, Minimum Distance Problem, Shortest Vector Problem}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.4
Improved (Provable) Algorithms for the Shortest Vector Problem via Bounded Distance Decoding

Authors: Divesh Aggarwal, Yanlin Chen, Rajendra Kumar, and Yixin Shen

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
The most important computational problem on lattices is the Shortest Vector Problem (SVP). In this paper, we present new algorithms that improve the state-of-the-art for provable classical/quantum algorithms for SVP. We present the following results. 1) A new algorithm for SVP that provides a smooth tradeoff between time complexity and memory requirement. For any positive integer 4 ≤ q ≤ √n, our algorithm takes q^{13n+o(n)} time and requires poly(n)⋅ q^{16n/q²} memory. This tradeoff which ranges from enumeration (q = √n) to sieving (q constant), is a consequence of a new time-memory tradeoff for Discrete Gaussian sampling above the smoothing parameter. 2) A quantum algorithm that runs in time 2^{0.9533n+o(n)} and requires 2^{0.5n+o(n)} classical memory and poly(n) qubits. This improves over the previously fastest classical (which is also the fastest quantum) algorithm due to [Divesh Aggarwal et al., 2015] that has a time and space complexity 2^{n+o(n)}. 3) A classical algorithm for SVP that runs in time 2^{1.741n+o(n)} time and 2^{0.5n+o(n)} space. This improves over an algorithm of [Yanlin Chen et al., 2018] that has the same space complexity. The time complexity of our classical and quantum algorithms are expressed using a quantity related to the kissing number of a lattice. A known upper bound of this quantity is 2^{0.402n}, but in practice for most lattices, it can be much smaller and even 2^o(n). In that case, our classical algorithm runs in time 2^{1.292n} and our quantum algorithm runs in time 2^{0.750n}.
Cite as

Divesh Aggarwal, Yanlin Chen, Rajendra Kumar, and Yixin Shen. Improved (Provable) Algorithms for the Shortest Vector Problem via Bounded Distance Decoding. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{aggarwal_et_al:LIPIcs.STACS.2021.4,
  author =	{Aggarwal, Divesh and Chen, Yanlin and Kumar, Rajendra and Shen, Yixin},
  title =	{{Improved (Provable) Algorithms for the Shortest Vector Problem via Bounded Distance Decoding}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{4:1--4:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.4},
  URN =		{urn:nbn:de:0030-drops-136494},
  doi =		{10.4230/LIPIcs.STACS.2021.4},
  annote =	{Keywords: Lattices, Shortest Vector Problem, Discrete Gaussian Sampling, Time-Space Tradeoff, Quantum computation, Bounded distance decoding}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.38
Hardness of Approximation of (Multi-)LCS over Small Alphabet

Authors: Amey Bhangale, Diptarka Chakraborty, and Rajendra Kumar

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

Abstract
The problem of finding longest common subsequence (LCS) is one of the fundamental problems in computer science, which finds application in fields such as computational biology, text processing, information retrieval, data compression etc. It is well known that (decision version of) the problem of finding the length of a LCS of an arbitrary number of input sequences (which we refer to as Multi-LCS problem) is NP-complete. Jiang and Li [SICOMP'95] showed that if Max-Clique is hard to approximate within a factor of s then Multi-LCS is also hard to approximate within a factor of Θ(s). By the NP-hardness of the problem of approximating Max-Clique by Zuckerman [ToC'07], for any constant δ > 0, the length of a LCS of arbitrary number of input sequences of length n each, cannot be approximated within an n^{1-δ}-factor in polynomial time unless {P}={NP}. However, the reduction of Jiang and Li assumes the alphabet size to be Ω(n). So far no hardness result is known for the problem of approximating Multi-LCS over sub-linear sized alphabet. On the other hand, it is easy to get 1/|Σ|-factor approximation for strings of alphabet Σ. In this paper, we make a significant progress towards proving hardness of approximation over small alphabet by showing a polynomial-time reduction from the well-studied densest k-subgraph problem with perfect completeness to approximating Multi-LCS over alphabet of size poly(n/k). As a consequence, from the known hardness result of densest k-subgraph problem (e.g. [Manurangsi, STOC'17]) we get that no polynomial-time algorithm can give an n^{-o(1)}-factor approximation of Multi-LCS over an alphabet of size n^{o(1)}, unless the Exponential Time Hypothesis is false.
Cite as

Amey Bhangale, Diptarka Chakraborty, and Rajendra Kumar. Hardness of Approximation of (Multi-)LCS over Small Alphabet. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bhangale_et_al:LIPIcs.APPROX/RANDOM.2020.38,
  author =	{Bhangale, Amey and Chakraborty, Diptarka and Kumar, Rajendra},
  title =	{{Hardness of Approximation of (Multi-)LCS over Small Alphabet}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{38:1--38:16},
  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.2020.38},
  URN =		{urn:nbn:de:0030-drops-126418},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.38},
  annote =	{Keywords: Longest common subsequence, Hardness of approximation, ETH-hardness, Densest k-subgraph problem}
}
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