Alphabet Reduction for Reconfiguration Problems

Abstract

We present a reconfiguration analogue of alphabet reduction \`a la Dinur (J. ACM, 2007) and its applications. Given a binary constraint graph $G$ and its two satisfying assignments $\psi^\mathsf{ini}$ and $\psi^\mathsf{tar}$, the Maxmin Binary CSP Reconfiguration problem requests to transform $\psi^\mathsf{ini}$ into $\psi^\mathsf{tar}$ 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 $\Omega(\varepsilon)$-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.