We study the Monotone Sliding Reconfiguration (MSR) problem, in which $\textit{labeled}$ pairwise interior-disjoint objects in a planar workspace need to be brought $\textit{one by one}$ from their initial positions to given target positions, without causing collisions. That is, at each step only one object moves to its respective target, where it stays thereafter. MSR is a natural special variant of Multi-Robot Motion Planning (MRMP) and related reconfiguration problems, many of which are known to be computationally hard. A key question is identifying the minimal mitigating assumptions that enable efficient algorithms for such problems. We first show that despite the monotonicity requirement, MSR remains a computationally hard MRMP problem. We then provide additional hardness results for MSR that rule out several natural assumptions. For example, we show that MSR remains hard without obstacles in the workspace. On the positive side, we introduce a family of MSR instances that always have a solution through a novel structural assumption pertaining to the graphs underlying the start and target configuration -- we require that these graphs are spannable by a forest of full binary trees (SFFBT). We use our assumption to obtain efficient MSR algorithms for unit discs and 2D grid settings. Notably, our assumption does not require separation between start/target positions, which is a standard requirement in efficient and complete MRMP algorithms. Instead, we (implicitly) require separation between $\textit{groups}$ of these positions, thereby pushing the boundary of efficiently solvable instances toward denser scenarios.

翻译：我们研究单调滑动重构（MSR）问题，该问题要求将平面工作空间中成对内部不相交的带标签物体逐个从其初始位置移动到给定目标位置，且不引发碰撞。即每一步仅有一个物体移动到其对应目标位置并保持静止。MSR是多机器人运动规划（MRMP）及相关重构问题的一个自然特例，而这类问题大多已被证明具有计算复杂性。关键问题在于识别能够使此类问题获得高效算法的最小缓解假设。我们首先证明，尽管存在单调性要求，MSR仍属于计算困难的MRMP问题。随后我们给出了MSR的附加困难性结果，排除了若干自然假设。例如，我们证明即使工作空间不存在障碍物，MSR仍保持困难性。在积极方面，我们通过引入关于起始与目标配置底层图结构的新颖假设——要求这些图可由满二叉树森林生成（SFFBT），构建了一类始终存在解的MSR实例族。基于该假设，我们为单位圆盘和二维网格场景设计了高效的MSR算法。值得注意的是，我们的假设不要求起始/目标位置间存在间距（这是高效完备MRMP算法的标准要求），而是（隐式地）要求这些位置的分组间存在间距，从而将高效可解实例的边界推向更密集的场景。