The field of combinatorial reconfiguration studies search problems with a focus on transforming one feasible solution into another. Recently, Ohsaka [STACS'23] put forth the Reconfiguration Inapproximability Hypothesis (RIH), which roughly asserts that for some $\epsilon>0$, 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 prove RIH and establish the first (constant factor) PSPACE-hardness of approximation results for many reconfiguration problems, resolving an open question posed by Ito et al. [TCS'11]. Our proof uses known constructions of Probabilistically Checkable Proofs of Proximity (in a black-box manner) to create the gap. Independently, Hirahara and Ohsaka [STOC'24] have also 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 $(1/2 - \epsilon)$-approximation polynomial time 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].

翻译：组合重配置领域研究搜索问题，重点关注将一个可行解转化为另一个可行解。最近，Ohsaka [STACS'23] 提出了重配置不可近似性假设（RIH），该假设大致断言：对于某个ϵ>0，给定一个k-CSP实例（k为常数，字母表规模为常数）及其两个满足性赋值ψ_s和ψ_t，寻找一个从ψ_s开始到ψ_t结束的赋值序列是PSPACE-困难的，其中序列中的每个赋值至少满足(1-ϵ)比例的约束，且每个赋值均通过改变其直接前驱赋值（在序列中）的恰好一个变量得到。假设RIH成立，Ohsaka [STACS'23; SODA'24] 已证明许多重要重配置问题的近似求解是PSPACE-困难的。本文证明了RIH，并首次建立了众多重配置问题的（常数因子）PSPACE-难度近似结果，解决了Ito等人 [TCS'11] 提出的一个开放问题。我们的证明采用已知的邻近概率可检验证明构造（以黑箱方式）来产生间隙。独立地，Hirahara和Ohsaka [STOC'24] 也证明了RIH。此外，我们还证明了当k=2时，上述k-CSP重配置问题在1/2+ϵ因子内（对任意ϵ>0）是NP-难度近似的。我们通过一个(1/2-ϵ)-近似多项式时间算法对此进行补充，该算法改进了Ohsaka [2023] 的(1/4-ϵ)-近似算法（同样针对任意ϵ>0）。最后，我们证明集合覆盖重配置问题在任意常数ϵ>0时，在2-ϵ因子内是NP-难度近似的，这与Ito等人 [TCS'11] 的简单线性时间2-近似算法相匹配。