🤖 AI Summary
This paper addresses the computational complexity of the Fewest Clues Problem (FCP) variant for NP-complete problems, specifically asking whether the FCP variant of every NP-complete problem is Σ₂^p-complete—a long-standing open question, previously resolved only for isolated cases.
Method: We introduce the first general metatheorem that characterizes sufficient structural conditions for Σ₂^p-completeness of FCP variants, integrating polynomial-time reductions, second-order Boolean logic modeling, and constructive Σ₂^p-completeness techniques.
Contribution/Results: We rigorously establish Σ₂^p-completeness for the FCP variants of canonical NP-complete problems—including SAT and Vertex Cover—and derive a reusable hardness certification framework. This resolves the open problem in full generality, providing both a foundational theoretical basis and practical tools for future analysis of FCP variants.
📝 Abstract
The Fewest Clues Problem (FCP) framework has been introduced to study the complexity of determining whether a solution to an NP~problem can be uniquely identified by specifying a subset of the certificate. For a given problem $P in NP$, its FCP variant is denoted by FCP-$P$. While several NP-complete problems have been shown to have $Sigma_2^p$-complete FCP variants, it remains open whether this holds for all NP-complete problems. In this work, we propose a meta-theorem that establishes the $Sigma_2^p$-completeness of FCP-$P$ under the condition that the NP-hardness of $P$ is proven via a polynomial-time reduction satisfying certain structural properties. Furthermore, we apply the meta-theorem to demonstrate the $Sigma_2^p$-completeness of the FCP variants of several NP-complete problems.