TY - GEN
T1 - Approximating Red-Blue Set Cover and Minimum Monotone Satisfying Assignment
AU - Chlamtáč, Eden
AU - Makarychev, Yury
AU - Vakilian, Ali
N1 - Publisher Copyright:
© 2023 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.
PY - 2023/9/1
Y1 - 2023/9/1
N2 - 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(m1/3)-approximation improving on the Õ(m1/2)-approximation due to Elkin and Peleg (where m is the number of sets). Our approximation algorithm for MMSAt (for circuits of depth t) gives an Õ(N1−δ) approximation for δ = 1323−⌈t/2⌉, where N is the number of gates and variables. No non-trivial approximation algorithms for MMSAt 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 Ω̃(m1/4−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 N1−ε where ε → 0 as the circuit depth t → ∞.
AB - 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(m1/3)-approximation improving on the Õ(m1/2)-approximation due to Elkin and Peleg (where m is the number of sets). Our approximation algorithm for MMSAt (for circuits of depth t) gives an Õ(N1−δ) approximation for δ = 1323−⌈t/2⌉, where N is the number of gates and variables. No non-trivial approximation algorithms for MMSAt 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 Ω̃(m1/4−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 N1−ε where ε → 0 as the circuit depth t → ∞.
KW - Circuit Minimum Monotone Satisfying Assignment (MMSA) Problem
KW - LP Rounding
KW - Red-Blue Set Cover Problem
UR - https://www.scopus.com/pages/publications/85171995290
U2 - 10.4230/LIPIcs.APPROX/RANDOM.2023.11
DO - 10.4230/LIPIcs.APPROX/RANDOM.2023.11
M3 - Conference contribution
AN - SCOPUS:85171995290
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023
A2 - Megow, Nicole
A2 - Smith, Adam
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023
Y2 - 11 September 2023 through 13 September 2023
ER -