Gap Amplification for Reconfiguration Problems

TL;DR

This work establishes explicit gap amplification results for reconfiguration problems under the Reconfiguration Inapproximability Hypothesis, showing that Maxmin 2-CSP Reconfiguration is PSPACE-hard to approximate within $0.9942$. The authors adapt Dinur's PCP gap amplification with expanderization and powering, augmented by alphabet squaring to preserve perfect completeness, and achieve a final inapproximability gap while maintaining strong expansion properties. They further translate these results into hardness for Minmax Set Cover Reconfiguration (hard to approximate within $1.0029$) and for Minmax Dominating Set Reconfiguration, and prove NP-hardness of approximation for Maxmin 2-CSP Reconfiguration at the $3/4$ threshold. The methods rely on gap-preserving reductions that use the expander mixing lemma and PCP-inspired verification, providing the first explicit inapproximability results for reconfiguration problems without parallel repetition and with unconditional variants discussed in follow-up work.

Abstract

In this paper, we demonstrate gap amplification for reconfiguration problems. In particular, we prove an explicit factor of PSPACE-hardness of approximation for three popular reconfiguration problems only assuming the Reconfiguration Inapproximability Hypothesis (RIH) due to Ohsaka (STACS 2023). Our main result is that under RIH, Maxmin 2-CSP Reconfiguration is PSPACE-hard to approximate within a factor of $0.9942$. Moreover, the same result holds even if the constraint graph is restricted to $(d,λ)$-expander for arbitrarily small $\fracλ{d}$. The crux of its proof is an alteration of the gap amplification technique due to Dinur (J. ACM, 2007), which amplifies the $1$ vs. $1-\varepsilon$ gap for arbitrarily small $\varepsilon \in (0,1)$ up to the $1$ vs. $1-0.0058$ gap. As an application of the main result, we demonstrate that Minmax Set Cover Reconfiguration and Minmax Dominating Set Reconfiguratio} are PSPACE-hard to approximate within a factor of $1.0029$ under RIH. Our proof is based on a gap-preserving reduction from Label Cover to Set Cover due to Lund and Yannakakis (J. ACM, 1994). Unlike Lund--Yannakakis' reduction, the expander mixing lemma is essential to use. We highlight that all results hold unconditionally as long as "PSPACE-hard" is replaced by "NP-hard," and are the first explicit inapproximability results for reconfiguration problems without resorting to the parallel repetition theorem. We finally complement the main result by showing that it is NP-hard to approximate Maxmin 2-CSP Reconfiguration within a factor better than $\frac{3}{4}$.

Figures (1)

• Figure 1: Running examples of the verifier when the original constraint graph $G=(V,E,\Sigma,\Pi)$ represents a $3$-Coloring instance; i.e., $\Sigma \coloneq \{\textcolor{red!60!black}{\mathtt{r}}, \textcolor{green!50!black}{\mathtt{g}}, \textcolor{blue!70!black}{\mathtt{b}}\}$ and $\pi_e \coloneq \Sigma^2 \setminus \{(\textcolor{red!60!black}{\mathtt{r}},\textcolor{red!60!black}{\mathtt{r}}), (\textcolor{green!50!black}{\mathtt{g}},\textcolor{green!50!black}{\mathtt{g}}), (\textcolor{blue!70!black}{\mathtt{b}},\textcolor{blue!70!black}{\mathtt{b}})\}$ for all $e \in E$. Here, for a proof $\psi' \colon V \to \Sigma'^{d^{\mathsf{R}+1}}$, the verifier selects a random walk from $x$ to $y$, and is about to perform the test of $\psi'(x)$ and $\psi'(y)$ at $e_i \coloneq (v,w)$.

Theorems & Definitions (54)

• Lemma 1.1: Gap amplification lemma; informal; see \ref{['lem:amp:pow']}
• Theorem 1.2: informal; see \ref{['thm:sc']}
• Theorem 1.3: informal; see \ref{['thm:NP-BCSPR']}
• Lemma 2.1
• Definition 2.2
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3: Gap amplification lemma
• Definition 3.4: guruswami2005course