On Inapproximability of Reconfiguration Problems: PSPACE-Hardness and some Tight NP-Hardness Results

TL;DR

This paper proves the Reconfiguration Inapproximability Hypothesis (RIH), establishing PSPACE-hardness for certain reconfiguration problems, and provides new NP-hardness and approximation results for k-CSP and Set Cover Reconfiguration problems.

Contribution

It offers the first proof of RIH using PCP of Proximity and parameterized inapproximability, and improves approximation bounds for k-CSP and Set Cover Reconfiguration.

Findings

Proved RIH assuming known constructions.

NP-hard to approximate 2-CSP Reconfiguration within 1/2 + ε.

Polynomial-time approximation algorithm with factor 1/2 - ε.

Abstract

Recently, Ohsaka [STACS'23] put forth the Reconfiguration Inapproximability Hypothesis (RIH), which roughly asserts that there is some $\epsilon>0$ such that given as input a $k$-CSP instance (for some constant $k$) over some constant sized alphabet, and two satisfying assignments $\psi_s$ and $\psi_t$, it is PSPACE-hard to find a sequence of assignments starting from $\psi_s$ and ending at $\psi_t$ such that every assignment in the sequence satisfies at least $(1-\epsilon)$ fraction of the constraints and also that every assignment in the sequence is obtained by changing its immediately preceding assignment (in the sequence) on exactly one variable. Assuming RIH, many important reconfiguration problems have been shown to be PSPACE-hard to approximate by Ohsaka [STACS'23; SODA'24]. In this paper, we provide a proof of RIH. Our proof uses known constructions of PCP of Proximity to create the gap, and further leverages a parallelization framework from recent parameterized inapproximability results to analyze the quantitative trade-off between $\epsilon$ and $k$ in RIH. We note that Hirahara and Ohsaka [STOC'24] have also independently proved RIH. We also prove that the aforementioned $k$-CSP Reconfiguration problem is NP-hard to approximate to within a factor of $1/2 + \epsilon$ (for any $\epsilon>0$) when $k=2$. We complement this with a polynomial time $(1/2 - \epsilon)$-approximation algorithm, which improves upon a $(1/4 - \epsilon)$-approximation algorithm of Ohsaka [2023] (again for any $\epsilon>0$). Finally, we show that Set Cover Reconfiguration is NP-hard to approximate to within a factor of $2 - \epsilon$ for any constant $\epsilon > 0$, which matches the simple linear-time 2-approximation algorithm by Ito et al. [TCS'11].