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Alphabet Reduction for Reconfiguration Problems

Naoto Ohsaka

TL;DR

This paper develops a reconfiguration analogue of alphabet reduction for Maxmin BCSP Reconfiguration, achieving a polynomial-time reduction from arbitrary alphabet sizes to a universal constant $W_0$ while preserving perfect completeness and reducing the gap by a universal factor $\kappa$. The core tools are the reconfigurability properties of Hadamard codes and a robustization framework that converts BCSP constraints into circuit SAT reconfigurations, followed by a composition step using assignment testers to obtain a small-alphabet, gap-preserving 4-CSP reconfiguration instance. By pairing robustization with gap amplification results, it establishes a gap-preserving reduction that is independent of the original gap parameter and thus enables PSPACE-hardness results under the Reconfiguration Inapproximability Hypothesis (RIH) with a universal alphabet size. The work thereby decouples alphabet blow-up from gap parameters and extends gap-preserving reductions to a wider class of reconfiguration problems, offering a foundation for Dinur-style PCP approaches in reconfiguration contexts. Open questions include extending reconfigurability to other codes and refining composition schemes for probabilistic reconfiguration proofs.

Abstract

We present a reconfiguration analogue of alphabet reduction à la Dinur (J. ACM, 2007) and its applications. Given a binary constraint graph $G$ and its two satisfying assignments $ψ^\mathsf{ini}$ and $ψ^\mathsf{tar}$, the Maxmin Binary CSP Reconfiguration problem requests to transform $ψ^\mathsf{ini}$ into $ψ^\mathsf{tar}$ by repeatedly changing the value of a single vertex so that the minimum fraction of satisfied edges is maximized. We demonstrate a polynomial-time reduction from Maxmin Binary CSP Reconfiguration with arbitrarily large alphabet size $W \in \mathbb{N}$ to itself with universal alphabet size $W_0 \in \mathbb{N}$ such that 1. the perfect completeness is preserved, and 2. if any reconfiguration for the former violates $\varepsilon$-fraction of edges, then $Ω(\varepsilon)$-fraction of edges must be unsatisfied during any reconfiguration for the latter. The crux of its construction is the reconfigurability of Hadamard codes, which enables to reconfigure between a pair of codewords, while avoiding getting too close to the other codewords. Combining this alphabet reduction with gap amplification due to Ohsaka (SODA 2024), we are able to amplify the $1$ vs. $1-\varepsilon$ gap for arbitrarily small $\varepsilon \in (0,1)$ up to the $1$ vs. $1-\varepsilon_0$ for some universal $\varepsilon_0 \in (0,1)$ without blowing up the alphabet size. In particular, a $1$ vs. $1-\varepsilon_0$ gap version of Maxmin Binary CSP Reconfiguration with alphabet size $W_0$ is PSPACE-hard only assuming the Reconfiguration Inapproximability Hypothesis posed by Ohsaka (STACS 2023), whose gap parameter can be arbitrarily small. This may not be achieved only by gap amplification of Ohsaka, which makes the alphabet size gigantic depending on the gap value of the hypothesis.

Alphabet Reduction for Reconfiguration Problems

TL;DR

This paper develops a reconfiguration analogue of alphabet reduction for Maxmin BCSP Reconfiguration, achieving a polynomial-time reduction from arbitrary alphabet sizes to a universal constant while preserving perfect completeness and reducing the gap by a universal factor . The core tools are the reconfigurability properties of Hadamard codes and a robustization framework that converts BCSP constraints into circuit SAT reconfigurations, followed by a composition step using assignment testers to obtain a small-alphabet, gap-preserving 4-CSP reconfiguration instance. By pairing robustization with gap amplification results, it establishes a gap-preserving reduction that is independent of the original gap parameter and thus enables PSPACE-hardness results under the Reconfiguration Inapproximability Hypothesis (RIH) with a universal alphabet size. The work thereby decouples alphabet blow-up from gap parameters and extends gap-preserving reductions to a wider class of reconfiguration problems, offering a foundation for Dinur-style PCP approaches in reconfiguration contexts. Open questions include extending reconfigurability to other codes and refining composition schemes for probabilistic reconfiguration proofs.

Abstract

We present a reconfiguration analogue of alphabet reduction à la Dinur (J. ACM, 2007) and its applications. Given a binary constraint graph and its two satisfying assignments and , the Maxmin Binary CSP Reconfiguration problem requests to transform into by repeatedly changing the value of a single vertex so that the minimum fraction of satisfied edges is maximized. We demonstrate a polynomial-time reduction from Maxmin Binary CSP Reconfiguration with arbitrarily large alphabet size to itself with universal alphabet size such that 1. the perfect completeness is preserved, and 2. if any reconfiguration for the former violates -fraction of edges, then -fraction of edges must be unsatisfied during any reconfiguration for the latter. The crux of its construction is the reconfigurability of Hadamard codes, which enables to reconfigure between a pair of codewords, while avoiding getting too close to the other codewords. Combining this alphabet reduction with gap amplification due to Ohsaka (SODA 2024), we are able to amplify the vs. gap for arbitrarily small up to the vs. for some universal without blowing up the alphabet size. In particular, a vs. gap version of Maxmin Binary CSP Reconfiguration with alphabet size is PSPACE-hard only assuming the Reconfiguration Inapproximability Hypothesis posed by Ohsaka (STACS 2023), whose gap parameter can be arbitrarily small. This may not be achieved only by gap amplification of Ohsaka, which makes the alphabet size gigantic depending on the gap value of the hypothesis.
Paper Structure (17 sections, 10 theorems, 37 equations, 2 figures, 2 tables)

This paper contains 17 sections, 10 theorems, 37 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Our Results
  4. Proof Overview
  5. Towards Dinur's Style Proof of RIH?
  6. Additional Related Work
  7. Preliminaries
  8. Notations
  9. Constraint Satisfaction Problem and Reconfigurability
  10. Alphabet Reduction for Maxmin BCSP Reconfiguration
  11. Reconfigurability of Hadamard Codes
  12. Robustization
  13. Reduction.
  14. Composition of Assignment Testers
  15. Reduction.
  16. ...and 2 more sections

Key Result

Theorem 1.1

There exist universal constants $W_0 \in \mathbb{N}$ and $\kappa \in (0,1)$ and a polynomial-time reduction from Maxmin Binary CSP Reconfiguration with arbitrarily large alphabet size $W \in \mathbb{N}$ to itself with alphabet size $W_0$ such that

Figures (2)

  • Figure 1: Illustration of $(P_\alpha, P_\beta, P_\gamma, P_=)$ for three distinct nonzero vectors $\mathbf{\bm{\alpha}}, \mathbf{\bm{\beta}}, \mathbf{\bm{\gamma}} \in \mathbb{F}_2^n$.
  • Figure 2: Plot of the distance from $f^{(t)}$ to $\mathop{\mathrm{\mathsf{Had}}}\nolimits(\mathbf{\bm{\alpha}})$, $\mathop{\mathrm{\mathsf{Had}}}\nolimits(\mathbf{\bm{\beta}})$, and $\mathop{\mathrm{\mathsf{Had}}}\nolimits(\mathbf{\bm{\gamma}})$ for a random reconfiguration $\Pi$ from $\mathop{\mathrm{\mathsf{Had}}}\nolimits(\mathbf{\bm{\alpha}})$ to $\mathop{\mathrm{\mathsf{Had}}}\nolimits(\mathbf{\bm{\beta}})$ described in the proof of \ref{['lem:Hadmard-reconf']}.

Theorems & Definitions (22)

  • Theorem 1.1: Alphabet reduction; informal; see \ref{['thm:ABC-reduct']}
  • Corollary 1.2: from \ref{['thm:ABC-reduct']} and ohsaka2023gapohsaka2024gap
  • Theorem 1.3: from \ref{['cor:ABC-reduct']} and ohsaka2023gapohsaka2024gap
  • Example 1.4: Failed attempt
  • Lemma 1.5: Reconfigurability of Hadamard codes; informal; see \ref{['lem:Hadmard-reconf']}
  • Definition 2.1
  • Theorem 3.1: Alphabet reduction
  • Lemma 3.2: Reconfigurability of Hadamard codes
  • proof
  • Lemma 3.4
  • ...and 12 more