On the Approximability of Parameterized Minimum Monotone Satisfying Assignment

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-MMSA₂ is the well-known k-SetCover problem, whose parameterized inapproximability lies in the polylog(n) regime. In contrast, k-MMSA₄ captures k-MinLabel, for which known lower bounds give poly(n) inapproximability. Sandwiched by k-MMSA₂ and k-MMSA₄, the inapproximability of k-MMSA₃ remained comparatively unexplored. 
In this paper, we give an FPT-time O(2^k log n)-approximation algorithm for k-MMSA₃, 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-MMSA₃ to k-MMSA₂. Thus, stronger inapproximability for k-MMSA₃ would imply new hardness for k-MMSA₂, 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-MMSA₄ 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.