On the Approximability of Parameterized Minimum Monotone Satisfying Assignment

2026-07-07

AI Summary

This work investigates the approximability of the parameterized Minimum Monotone Satisfying Assignment problem at level three (k-MMSA₃), bridging the theoretical gap between k-MMSA₂ and k-MMSA₄. The authors present the first FPT approximation algorithm for k-MMSA₃ with a factor of O(2ᵏ log n) and establish a gap-preserving reduction to k-SetCover, revealing a sharp jump in approximation hardness across the MMSA hierarchy. Under the assumptions that W[2] ≠ FPT and the Exponential Time Hypothesis (ETH), they prove that k-MMSA₄ admits no nᵒ⁽¹⁾-factor FPT approximation and no nᴼ⁽¹/ᵏ⁾-factor approximation in time nᵒ⁽ᵏ⁾. These results precisely delineate a threshold where k-MMSA₃ remains approximable while k-MMSA₄ becomes inapproximable, offering a new avenue for surpassing known approximation lower bounds for k-MMSA₂.

Abstract

The parameterized Minimum Monotone Satisfying Assignment ($k$-MMSA) problem asks whether a monotone Boolean circuit admits a satisfying assignment of Hamming weight at most $k$. The MMSA hierarchy is defined by allowing a bounded number of alternations between AND and OR gates in the circuit. While the polynomial-time approximability of the MMSA hierarchy has been studied extensively, much less is known in the parameterized setting. In particular, $k$-MMSA$_2$ is the well-known $k$-SetCover problem, whose parameterized inapproximability lies in the $\text{polylog}(n)$ regime. In contrast, $k$-MMSA$_4$ captures $k$-MinLabel, for which known lower bounds give $\text{poly}(n)$ inapproximability. Sandwiched by $k$-MMSA$_2$ and $k$-MMSA$_4$, the inapproximability of $k$-MMSA$_3$ remained comparatively unexplored. In this paper, we give an FPT-time $O(2^k \log n)$-approximation algorithm for $k$-MMSA$_3$, suggesting that in the fixed-parameter regime, the third level of MMSA remains surprisingly close to the second level. Complementing this algorithm, we also give an FPT-time gap-preserving reduction from $k$-MMSA$_3$ to $k$-MMSA$_2$. Thus, stronger inapproximability for $k$-MMSA$_3$ would imply new hardness for $k$-MMSA$_2$, potentially offering a route around the current barriers for the latter problem. Revisiting Marx's reduction from $k$-MMSA$_t$ to gap $k$-MMSA$_{t+2}$, we also show that $k$-MMSA$_4$ admits no $n^{o(1)}$-factor FPT approximation unless W[2]=FPT, and no $n^{O(1/k)}$-factor approximation running in $n^{o(k)}$ time under ETH. These results separate the parameterized approximability behavior of the third and fourth levels and clarify where stronger inapproximability enters the $k$-MMSA hierarchy.

Problem

• parameterized complexity
• minimum monotone satisfying assignment
• inapproximability
• FPT approximation
• Boolean circuits

Innovation

• parameterized approximation
• monotone Boolean circuits
• MMSA hierarchy
• FPT reduction
• inapproximability