2 Search Results for "Singer, Michael"

Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.38
On Sketching Approximations for Symmetric Boolean CSPs

Authors: Joanna Boyland, Michael Hwang, Tarun Prasad, Noah Singer, and Santhoshini Velusamy

Published in: LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Abstract
A Boolean maximum constraint satisfaction problem, Max-CSP(f), is specified by a predicate f:{-1,1}^k → {0,1}. An n-variable instance of Max-CSP(f) consists of a list of constraints, each of which applies f to k distinct literals drawn from the n variables. For k = 2, Chou, Golovnev, and Velusamy [Chou et al., 2020] obtained explicit ratios characterizing the √ n-space streaming approximability of every predicate. For k ≥ 3, Chou, Golovnev, Sudan, and Velusamy [Chou et al., 2022] proved a general dichotomy theorem for √ n-space sketching algorithms: For every f, there exists α(f) ∈ (0,1] such that for every ε > 0, Max-CSP(f) is (α(f)-ε)-approximable by an O(log n)-space linear sketching algorithm, but (α(f)+ε)-approximation sketching algorithms require Ω(√n) space. In this work, we give closed-form expressions for the sketching approximation ratios of multiple families of symmetric Boolean functions. Letting α'_k = 2^{-(k-1)} (1-k^{-2})^{(k-1)/2}, we show that for odd k ≥ 3, α(kAND) = α'_k, and for even k ≥ 2, α(kAND) = 2α'_{k+1}. Thus, for every k, kAND can be (2-o(1))2^{-k}-approximated by O(log n)-space sketching algorithms; we contrast this with a lower bound of Chou, Golovnev, Sudan, Velingker, and Velusamy [Chou et al., 2022] implying that streaming (2+ε)2^{-k}-approximations require Ω(n) space! We also resolve the ratio for the "at-least-(k-1)-1’s" function for all even k; the "exactly-(k+1)/2-1’s" function for odd k ∈ {3,…,51}; and fifteen other functions. We stress here that for general f, the dichotomy theorem in [Chou et al., 2022] only implies that α(f) can be computed to arbitrary precision in PSPACE, and thus closed-form expressions need not have existed a priori. Our analyses involve identifying and exploiting structural "saddle-point" properties of this dichotomy. Separately, for all threshold functions, we give optimal "bias-based" approximation algorithms generalizing [Chou et al., 2020] while simplifying [Chou et al., 2022]. Finally, we investigate the √ n-space streaming lower bounds in [Chou et al., 2022], and show that they are incomplete for 3AND, i.e., they fail to rule out (α(3AND})-ε)-approximations in o(√ n) space.
Cite as

Joanna Boyland, Michael Hwang, Tarun Prasad, Noah Singer, and Santhoshini Velusamy. On Sketching Approximations for Symmetric Boolean CSPs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 38:1-38:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{boyland_et_al:LIPIcs.APPROX/RANDOM.2022.38,
  author =	{Boyland, Joanna and Hwang, Michael and Prasad, Tarun and Singer, Noah and Velusamy, Santhoshini},
  title =	{{On Sketching Approximations for Symmetric Boolean CSPs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{38:1--38:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.38},
  URN =		{urn:nbn:de:0030-drops-171604},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.38},
  annote =	{Keywords: Streaming algorithms, constraint satisfaction problems, approximability}
}
Document
DOI: 10.4230/DagSemRep.26
Efficient Interpolation Algorithms (Dagstuhl Seminar 9149)

Authors: Andreas Dress, Marek Karpinski, and Michael Singer

Published in: Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

Abstract
Cite as

Andreas Dress, Marek Karpinski, and Michael Singer. Efficient Interpolation Algorithms (Dagstuhl Seminar 9149). Dagstuhl Seminar Report 26, pp. 1-15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (1992)

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@TechReport{dress_et_al:DagSemRep.26,
  author =	{Dress, Andreas and Karpinski, Marek and Singer, Michael},
  title =	{{Efficient Interpolation Algorithms (Dagstuhl Seminar 9149)}},
  pages =	{1--15},
  ISSN =	{1619-0203},
  year =	{1992},
  type = 	{Dagstuhl Seminar Report},
  number =	{26},
  institution =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemRep.26},
  URN =		{urn:nbn:de:0030-drops-149141},
  doi =		{10.4230/DagSemRep.26},
}
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