License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.11
URN: urn:nbn:de:0030-drops-188366
URL: https://drops.dagstuhl.de/opus/volltexte/2023/18836/
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Chlamtáč, Eden ; Makarychev, Yury ; Vakilian, Ali

Approximating Red-Blue Set Cover and Minimum Monotone Satisfying Assignment

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LIPIcs-APPROX11.pdf (0.8 MB)


Abstract

We provide new approximation algorithms for the Red-Blue Set Cover and Circuit Minimum Monotone Satisfying Assignment (MMSA) problems. Our algorithm for Red-Blue Set Cover achieves Õ(m^{1/3})-approximation improving on the Õ(m^{1/2})-approximation due to Elkin and Peleg (where m is the number of sets). Our approximation algorithm for MMSA_t (for circuits of depth t) gives an Õ(N^{1-δ}) approximation for δ = 1/32^{3-⌈t/2⌉}, where N is the number of gates and variables. No non-trivial approximation algorithms for MMSA_t with t ≥ 4 were previously known.
We complement these results with lower bounds for these problems: For Red-Blue Set Cover, we provide a nearly approximation preserving reduction from Min k-Union that gives an ̃Ω(m^{1/4 - ε}) hardness under the Dense-vs-Random conjecture, while for MMSA we sketch a proof that an SDP relaxation strengthened by Sherali-Adams has an integrality gap of N^{1-ε} where ε → 0 as the circuit depth t → ∞.

BibTeX - Entry

@InProceedings{chlamtac_et_al:LIPIcs.APPROX/RANDOM.2023.11,
  author =	{Chlamt\'{a}\v{c}, Eden and Makarychev, Yury and Vakilian, Ali},
  title =	{{Approximating Red-Blue Set Cover and Minimum Monotone Satisfying Assignment}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{11:1--11:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18836},
  URN =		{urn:nbn:de:0030-drops-188366},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.11},
  annote =	{Keywords: Red-Blue Set Cover Problem, Circuit Minimum Monotone Satisfying Assignment (MMSA) Problem, LP Rounding}
}

Keywords: Red-Blue Set Cover Problem, Circuit Minimum Monotone Satisfying Assignment (MMSA) Problem, LP Rounding
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)
Issue Date: 2023
Date of publication: 04.09.2023


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