Completely inapproximable monotone and antimonotone parameterized problems

TL;DR

This paper proves that certain weighted circuit satisfiability problems for monotone and antimonotone circuits are completely inapproximable within fixed-parameter tractability, unless a major complexity class separation fails.

Contribution

It establishes the inapproximability of weighted circuit satisfiability for monotone and antimonotone circuits under standard complexity assumptions.

Findings

No fixed-parameter tractable approximation algorithms exist for these problems.

Such problems cannot be approximated within any nontrivial ratio in polynomial time.

The results imply strong inapproximability even for classical approximation algorithms.

Abstract

We prove that weighted circuit satisfiability for monotone or antimonotone circuits has no fixed-parameter tractable approximation algorithm with any approximation ratio function $\rho$, unless $FPT\neq W[1]$. In particular, not having such an fpt-approximation algorithm implies that these problems have no polynomial-time approximation algorithms with ratio $\rho(OPT)$ for any nontrivial function $\rho$.