Approximating Red-Blue Set Cover and Minimum Monotone Satisfying Assignment

Authors Eden Chlamtáč , Yury Makarychev , Ali Vakilian

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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

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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)


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
  • Red-Blue Set Cover Problem
  • Circuit Minimum Monotone Satisfying Assignment (MMSA) Problem
  • LP Rounding


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  1. V. P. Abidha and Pradeesha Ashok. Red blue set cover problem on axis-parallel hyperplanes and other objects. CoRR, abs/2209.06661, 2022. URL:
  2. Michael Alekhnovich, Samuel R. Buss, Shlomo Moran, and Toniann Pitassi. Minimum propositional proof length is np-hard to linearly approximate. J. Symb. Log., 66(1):171-191, 2001. URL:
  3. Pradeesha Ashok, Sudeshna Kolay, and Saket Saurabh. Multivariate complexity analysis of geometric red blue set cover. Algorithmica, 79(3):667-697, 2017. URL:
  4. Pranjal Awasthi, Avrim Blum, and Or Sheffet. Improved guarantees for agnostic learning of disjunctions. In Adam Tauman Kalai and Mehryar Mohri, editors, COLT 2010 - The 23rd Conference on Learning Theory, Haifa, Israel, June 27-29, 2010, pages 359-367. Omnipress, 2010. URL:
  5. Robert D. Carr, Srinivas Doddi, Goran Konjevod, and Madhav V. Marathe. On the red-blue set cover problem. In David B. Shmoys, editor, Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, January 9-11, 2000, San Francisco, CA, USA, pages 345-353. ACM/SIAM, 2000. URL:
  6. Timothy M. Chan and Nan Hu. Geometric red-blue set cover for unit squares and related problems. Comput. Geom., 48(5):380-385, 2015. URL:
  7. Moses Charikar, Yonatan Naamad, and Anthony Wirth. On approximating target set selection. In Klaus Jansen, Claire Mathieu, José D. P. Rolim, and Chris Umans, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2016, September 7-9, 2016, Paris, France, volume 60 of LIPIcs, pages 4:1-4:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL:
  8. Eden Chlamtác, Michael Dinitz, Christian Konrad, Guy Kortsarz, and George Rabanca. The densest k-subhypergraph problem. SIAM J. Discret. Math., 32(2):1458-1477, 2018. URL:
  9. Eden Chlamtác, Michael Dinitz, and Yury Makarychev. Minimizing the union: Tight approximations for small set bipartite vertex expansion. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 881-899. SIAM, 2017. URL:
  10. Irit Dinur and Shmuel Safra. On the hardness of approximating label-cover. Inf. Process. Lett., 89(5):247-254, 2004. URL:
  11. Michael Elkin and David Peleg. The hardness of approximating spanner problems. Theory Comput. Syst., 41(4):691-729, 2007. URL:
  12. Michael H. Goldwasser and Rajeev Motwani. Intractability of assembly sequencing: Unit disks in the plane. In Frank K. H. A. Dehne, Andrew Rau-Chaplin, Jörg-Rüdiger Sack, and Roberto Tamassia, editors, Algorithms and Data Structures, 5th International Workshop, WADS '97, Halifax, Nova Scotia, Canada, August 6-8, 1997, Proceedings, volume 1272 of Lecture Notes in Computer Science, pages 307-320. Springer, 1997. URL:
  13. Raghunath Reddy Madireddy and Apurva Mudgal. A constant-factor approximation algorithm for red-blue set cover with unit disks. Algorithmica, 85(1):100-132, 2023. URL:
  14. Raghunath Reddy Madireddy, Subhas C. Nandy, and Supantha Pandit. On the geometric red-blue set cover problem. In Ryuhei Uehara, Seok-Hee Hong, and Subhas C. Nandy, editors, WALCOM: Algorithms and Computation - 15th International Conference and Workshops, WALCOM 2021, Yangon, Myanmar, February 28 - March 2, 2021, Proceedings, volume 12635 of Lecture Notes in Computer Science, pages 129-141. Springer, 2021. URL:
  15. Pauli Miettinen. On the positive-negative partial set cover problem. Inf. Process. Lett., 108(4):219-221, 2008. URL: