3 Search Results for "Chawin, Dror"

Document
DOI: 10.4230/LIPIcs.MFCS.2025.37
New Hardness Results for Low-Rank Matrix Completion

Authors: Dror Chawin and Ishay Haviv

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
The low-rank matrix completion problem asks whether a given real matrix with missing values can be completed so that the resulting matrix has low rank or is close to a low-rank matrix. The completed matrix is often required to satisfy additional structural constraints, such as positive semi-definiteness or a bounded infinity norm. The problem arises in various research fields, including machine learning, statistics, and theoretical computer science, and has broad real-world applications. This paper presents new NP-hardness results for low-rank matrix completion problems. We show that for every sufficiently large integer d and any real number ε ∈ [2^{-O(d)},1/7], given a partial matrix A with exposed values of magnitude at most 1 that admits a positive semi-definite completion of rank d, it is NP-hard to find a positive semi-definite matrix that agrees with each given value of A up to an additive error of at most ε, even when the rank is allowed to exceed d by a multiplicative factor of O (1/(ε²⋅log(1/ε))). This strengthens a result of Hardt, Meka, Raghavendra, and Weitz (COLT, 2014), which applies to multiplicative factors smaller than 2 and to ε that decays polynomially in d. We establish similar NP-hardness results for the case where the completed matrix is constrained to have a bounded infinity norm (rather than be positive semi-definite), for which all previous hardness results rely on complexity assumptions related to the Unique Games Conjecture. Our proofs involve a novel notion of nearly orthonormal representations of graphs, the concept of line digraphs, and bounds on the rank of perturbed identity matrices.
Cite as

Dror Chawin and Ishay Haviv. New Hardness Results for Low-Rank Matrix Completion. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chawin_et_al:LIPIcs.MFCS.2025.37,
  author =	{Chawin, Dror and Haviv, Ishay},
  title =	{{New Hardness Results for Low-Rank Matrix Completion}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{37:1--37:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.37},
  URN =		{urn:nbn:de:0030-drops-241448},
  doi =		{10.4230/LIPIcs.MFCS.2025.37},
  annote =	{Keywords: hardness of approximation, low-rank matrix completion, graph coloring}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.39
Nearly Orthogonal Sets over Finite Fields

Authors: Dror Chawin and Ishay Haviv

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
For a field 𝔽 and integers d and k, a set of vectors of 𝔽^d is called k-nearly orthogonal if its members are non-self-orthogonal and every k+1 of them include an orthogonal pair. We prove that for every prime p there exists a positive constant δ = δ (p), such that for every field 𝔽 of characteristic p and for all integers k ≥ 2 and d ≥ k^{1/(p-1)}, there exists a k-nearly orthogonal set of at least d^{δ ⋅ k^{1/(p-1)} / log k} vectors of 𝔽^d. In particular, for the binary field we obtain a set of d^Ω(k/log k) vectors, and this is tight up to the log k term in the exponent. For comparison, the best known lower bound over the reals is d^Ω(log k / log log k)} (Alon and Szegedy, Graphs and Combin., 1999). The proof combines probabilistic and spectral arguments.
Cite as

Dror Chawin and Ishay Haviv. Nearly Orthogonal Sets over Finite Fields. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 39:1-39:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chawin_et_al:LIPIcs.SoCG.2024.39,
  author =	{Chawin, Dror and Haviv, Ishay},
  title =	{{Nearly Orthogonal Sets over Finite Fields}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{39:1--39:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.39},
  URN =		{urn:nbn:de:0030-drops-199848},
  doi =		{10.4230/LIPIcs.SoCG.2024.39},
  annote =	{Keywords: Nearly orthogonal sets, Finite fields}
}
Document
DOI: 10.4230/LIPIcs.STACS.2023.20
Improved NP-Hardness of Approximation for Orthogonality Dimension and Minrank

Authors: Dror Chawin and Ishay Haviv

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract
The orthogonality dimension of a graph G over ℝ is the smallest integer k for which one can assign a nonzero k-dimensional real vector to each vertex of G, such that every two adjacent vertices receive orthogonal vectors. We prove that for every sufficiently large integer k, it is NP-hard to decide whether the orthogonality dimension of a given graph over ℝ is at most k or at least 2^{(1-o(1))⋅k/2}. We further prove such hardness results for the orthogonality dimension over finite fields as well as for the closely related minrank parameter, which is motivated by the index coding problem in information theory. This in particular implies that it is NP-hard to approximate these graph quantities to within any constant factor. Previously, the hardness of approximation was known to hold either assuming certain variants of the Unique Games Conjecture or for approximation factors smaller than 3/2. The proofs involve the concept of line digraphs and bounds on their orthogonality dimension and on the minrank of their complement.
Cite as

Dror Chawin and Ishay Haviv. Improved NP-Hardness of Approximation for Orthogonality Dimension and Minrank. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chawin_et_al:LIPIcs.STACS.2023.20,
  author =	{Chawin, Dror and Haviv, Ishay},
  title =	{{Improved NP-Hardness of Approximation for Orthogonality Dimension and Minrank}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{20:1--20:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.20},
  URN =		{urn:nbn:de:0030-drops-176724},
  doi =		{10.4230/LIPIcs.STACS.2023.20},
  annote =	{Keywords: hardness of approximation, graph coloring, orthogonality dimension, minrank, index coding}
}
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