Tractability Frontiers in Multi-Robot Coordination and Geometric Reconfiguration

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.