LIPIcs.APPROX-RANDOM.2023.11.pdf
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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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Volume:
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Conference on Approximation Algorithms for Combinatorial Optimization Problems (APPROX)
Conference: International Conference on Randomization and Computation (RANDOM) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2023-09-04
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Author Details
- Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Cite AsGet BibTex
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
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.11
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 → ∞.
Subject Classification
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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References
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