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Approximating Red-Blue Set Cover and Minimum Monotone Satisfying Assignment

Authors Eden Chlamtáč , Yury Makarychev , Ali Vakilian

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Circuit complexity
Keywords
  • Red-Blue Set Cover Problem
  • Circuit Minimum Monotone Satisfying Assignment (MMSA) Problem
  • LP Rounding

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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 → ∞.

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Eden Chlamtáč, Yury Makarychev, and Ali Vakilian. Approximating Red-Blue Set Cover and Minimum Monotone Satisfying Assignment. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.11

Author Details

Eden Chlamtáč
  • Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Yury Makarychev
  • Toyota Technological Institute at Chicago (TTIC), IL, USA
Ali Vakilian
  • Toyota Technological Institute at Chicago (TTIC), IL, USA

Funding

  • Chlamtáč, Eden: The work was done while the author was visiting and supported by TTIC.
  • Makarychev, Yury: Supported by NSF awards CCF-1955173 and CCF-1934843.

References

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