Given a two-prover game $G$ and its two satisfying labelings $\psi_\mathsf{s}$ and $\psi_\mathsf{t}$, the Label Cover Reconfiguration problem asks whether $\psi_\mathsf{s}$ can be transformed into $\psi_\mathsf{t}$ by repeatedly changing the value of a vertex while preserving any intermediate labeling satisfying $G$. We consider an optimization variant of Label Cover Reconfiguration by relaxing the feasibility of labelings, referred to as Maxmin Label Cover Reconfiguration: we are allowed to transform by passing through any non-satisfying labelings, but required to maximize the minimum fraction of satisfied edges during transformation from $\psi_\mathsf{s}$ to $\psi_\mathsf{t}$. Since the parallel repetition theorem of Raz (SIAM J. Comput., 1998), which implies NP-hardness of Label Cover within any constant factor, produces strong inapproximability results for many NP-hard problems, one may think of using Maxmin Label Cover Reconfiguration to derive inapproximability results for reconfiguration problems. We prove the following results on Maxmin Label Cover Reconfiguration, which display different trends from those of Label Cover and the parallel repetition theorem: (1) Maxmin Label Cover Reconfiguration can be approximated within a factor of nearly $\frac{1}{4}$ for restricted graph classes, including slightly dense graphs and balanced bipartite graphs. (2) A naive parallel repetition of Maxmin Label Cover Reconfiguration does not decrease the optimal objective value. (3) Label Cover Reconfiguration on projection games can be decided in polynomial time. The above results suggest that a reconfiguration analogue of the parallel repetition theorem is unlikely.

翻译：给定一个双证明者游戏$G$及其两个满足标签赋值$\psi_\mathsf{s}$和$\psi_\mathsf{t}$，标签覆盖重构问题询问是否可以通过反复改变某个顶点的取值，同时保持所有中间赋值均满足$G$，将$\psi_\mathsf{s}$转化为$\psi_\mathsf{t}$。我们通过放宽赋值的可行性，考虑标签覆盖重构的一个优化变体，称为最大最小标签覆盖重构：允许通过任何不满足的赋值进行变换，但要求在从$\psi_\mathsf{s}$到$\psi_\mathsf{t}$的变换过程中最大化满足边的最小比例。由于Raz的并行重复定理（SIAM J. Comput., 1998）表明标签覆盖在任何常数因子内均为NP难问题，这为许多NP难问题提供了强不可近似性结果，因此人们可能考虑利用最大最小标签覆盖重构来推导重构问题的不可近似性结果。我们证明关于最大最小标签覆盖重构的如下结果，这些结果显示出与标签覆盖及并行重复定理不同的趋势：（1）对于受限图类（包括略稠密图和平衡二分图），最大最小标签覆盖重构可在接近$\frac{1}{4}$的因子内近似。（2）对最大最小标签覆盖重构进行简单的并行重复不会降低最优目标值。（3）投影游戏上的标签覆盖重构可在多项式时间内判定。上述结果表明，并行重复定理的重构类比不太可能存在。