Approximating Red-Blue Set Cover and Minimum Monotone Satisfying Assignment

TL;DR

This paper introduces improved approximation algorithms for Red-Blue Set Cover and Minimum Monotone Satisfying Assignment problems, along with new lower bounds and hardness results, advancing the understanding of these computational challenges.

Contribution

It presents the first improved approximation algorithms for MMSA$_t$ with circuit depth $t eq 2$, and enhances the approximation for Red-Blue Set Cover, also providing new hardness and integrality gap results.

Findings

Red-Blue Set Cover approximation improved to $ ilde O(m^{1/3})$

MMSA$_t$ approximation achieved with $ ilde O(N^{1-rac{1}{3}2^{3-rac{t}{2}}})$

Lower bounds established via reductions and integrality gap analysis

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 $\tilde O(m^{1/3})$-approximation improving on the $\tilde O(m^{1/2})$-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 $\tilde O(N^{1-\delta})$ approximation for $\delta = \frac{1}{3}2^{3-\lceil t/2\rceil}$, where $N$ is the number of gates and variables. No non-trivial approximation algorithms for MMSA$_t$ with $t\geq 4$ 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 $\tilde\Omega(m^{1/4 - \varepsilon})$ 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 $N^{1-\varepsilon}$ where $\varepsilon \to 0$ as the circuit depth $t\to \infty$.