Tzvika Geft

Tzvika Geft

Multi-Agent Path Finding (MAPF) is a fundamental motion coordination problem arising in multi-agent systems with a wide range of applications. The problem's intractability has led to extensive research on improving the scalability of solvers for it. Since optimal solvers can struggle to scale, a major challenge that arises is understanding what makes MAPF hard. We tackle this challenge through a fine-grained complexity analysis of time-optimal MAPF on 2D grids, thereby closing two gaps and identifying a new tractability frontier. First, we show that 2-colored MAPF, i.e., where the agents are divided into two teams, each with its own set of targets, remains NP-hard. Second, for the flowtime objective (also called sum-of-costs), we show that it remains NP-hard to find a solution in which agents have an individually optimal cost, which we call an individually optimal solution. The previously tightest results for these MAPF variants are for (non-grid) planar graphs. We use a single hardness construction that replaces, strengthens, and unifies previous proofs. We believe that it is also simpler than previous proofs for the planar case as it employs minimal gadgets that enable its full visualization in one figure. Finally, for the flowtime objective, we establish a tractability frontier based on the number of directions agents can move in. Namely, we complement our hardness result, which holds for three directions, with an efficient algorithm for finding an individually optimal solution if only two directions are allowed. This result sheds new light on the structure of optimal solutions, which may help guide algorithm design for the general problem.

Tzvika Geft, Dan Halperin, Yonatan Nakar

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.

Pankaj K. Agarwal, Boris Aronov, Tzvika Geft et al.

Assembly planning is a fundamental problem in robotics and automation, which involves designing a sequence of motions to bring the separate constituent parts of a product into their final placement in the product. Assembly planning is naturally cast as a disassembly problem, giving rise to the assembly partitioning problem: Given a set $A$ of parts, find a subset $S\subset A$, referred to as a subassembly, such that $S$ can be rigidly translated to infinity along a prescribed direction without colliding with $A\setminus S$. While assembly partitioning is efficiently solvable, it is further desirable for the parts of a subassembly to be easily held together. This motivates the problem that we study, called connected-assembly-partitioning, which additionally requires each of the two subassemblies, $S$ and $A\setminus S$, to be connected. We show that this problem is NP-complete, settling an open question posed by Wilson et al. (1995) a quarter of a century ago, even when $A$ consists of unit-grid squares (i.e., $A$ is polyomino-shaped). Towards this result, we prove the NP-hardness of a new Planar 3-SAT variant having an adjacency requirement for variables appearing in the same clause, which may be of independent interest. On the positive side, we give an $O(2^k n^2)$-time fixed-parameter tractable algorithm (requiring low degree polynomial-time pre-processing) for an assembly $A$ consisting of polygons in the plane, where $n=|A|$ and $k=|S|$. We also describe a special case of unit-grid square assemblies, where a connected partition can always be found in $O(n)$-time.

Tzvika Geft, Aviv Tamar, Ken Goldberg et al.

To compute robust 2D assembly plans, we present an approach that combines geometric planning with a deep neural network. We train the network using the Box2D physics simulator with added stochastic noise to yield robustness scores--the success probabilities of planned assembly motions. As running a simulation for every assembly motion is impractical, we train a convolutional neural network to map assembly operations, given as an image pair of the subassemblies before and after they are mated, to a robustness score. The neural network prediction is used within a planner to quickly prune out motions that are not robust. We demonstrate this approach on two-handed planar assemblies, where the motions are one-step translations. Results suggest that the neural network can learn robustness to plan robust sequences an order of magnitude faster than physics simulation.