Meta Theorem for Hardness on FCP-Problem

TL;DR

The paper addresses the challenge of proving $Σ_2^P$-completeness for Fewest Clues Problem (FCP) variants of NP-complete problems. It introduces a meta-theorem that, given a problem $P$ with $FCP$-$P$ in $Σ_2^P$ and a parsimonious reduction to a problem $Q$ with a structured certificate mapping, guarantees $FCP$-$Q$ is also $Σ_2^P$-complete; this relies on per-coordinate injective or constant dependencies between certificate components. The main contributions are the formal meta-theorem, its proof framework, and its application to derive $Σ_2^P$-completeness for several FCP variants (e.g., Yosenabe, Choco Banana, Nondango, Double Choco). The results provide a general methodology to transfer $Σ_2^P$-hardness across FCP problems via certificate-structure-preserving parsimonious reductions, offering a principled route to extend Σ2-completeness beyond ad hoc proofs and informing future relaxation of the structural conditions.

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 $Σ_2^\p$-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 $Σ_2^\p$-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 $Σ_2^\p$-completeness of the FCP variants of several \NP-complete problems.

Figures (8)

• Figure 1: Overview of variables related to FCP-$P$ and FCP-$Q$ in Theorem \ref{['thm:meta']}. The conditions and relationships among $c_P$, $c_Q$, and others are assumptions of the theorem. The conclusion is that $c_Q$ exists and can uniquify $S_Q$ using $c_P$ and the parsimonious reduction.
• Figure 2: An example of $x_P$ and $x_Q$ satisfying the conditions of Theorem \ref{['thm:meta']}. Each character of $x_Q$ depends on exactly one character of $x_P$ via an injective function $f_j$.
• Figure 3: An example of $x_P$ and $x_Q$ that does not satisfy the conditions of Theorem \ref{['thm:meta']}. Here, $x_Q[2]$ depends on both $x_P[0]$ and $x_P[1]$, violating the uniqueness condition.
• Figure 4: Given a clue $c_P \subset x_P$ as shown, the set of specified indices is $L = \{0, 2, 3\}$.
• Figure 5: Initialization of clue $c_Q$: fill all entries with $\bot$.

Theorems & Definitions (17)

• Definition 1: Erik2018
• Definition 2: Erik2018
• Definition 3
• Theorem 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Corollary 7