On Two-Handed Planar Assembly Partitioning with Connectivity Constraints (2009.12369v2)

Abstract: 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.

• The paper proves that connected assembly partitioning is NP-complete, solving a long-standing open problem in computational geometry.
• The paper provides a fixed-parameter tractable algorithm efficient for small subassemblies and a linear-time algorithm for horizontally monotone assemblies.
• These results offer theoretical insights and practical algorithms for optimizing design and automation processes in manufacturing.

Insights on "On Two-Handed Planar Assembly Partitioning with Connectivity Constraints"

The paper, "On Two-Handed Planar Assembly Partitioning with Connectivity Constraints" by Agarwal, Aronov, Gefts, and Halperins, explores the intricate problem of assembly planning in robotics and automation. This research focuses on the connected-assembly-partitioning problem, introducing complexity and innovative approaches to an issue that plays a significant role in optimizing manufacturing processes.

Key Contributions

A fundamental contribution of this paper is the proof that the connected-assembly-partitioning problem is NP-complete, even when the assembly is represented as a set of unit-grid squares. This resolves a longstanding open question about the complexity of connected assembly partitioning with a single translation, posed more than two decades ago by Wilson et al. The authors accomplish this by demonstrating the NP-hardness of a novel Planar 3-SAT variant with adjacency constraints for variables in the same clause. This theoretical advancement not only strengthens the understanding of geometric and combinatorial problems associated with assembly planning but may also draw interest due to its implications on other computational problems.

Moreover, the authors present a fixed-parameter tractable (FPT) algorithm for the problem, efficient for small subassemblies, with a time complexity of O(2^kn^2), where n represents the number of parts and k the size of the subassembly. This algorithm employs a bounded search tree approach to optimize partitioning and is applicable after a low-degree polynomial-time preprocessing. The specialized algorithm provides both practical and theoretical contributions by reducing the reliance on exponentially complex exhaustive strategies that were previously common.

Furthermore, a linear-time algorithm is offered for a subset of unit-grid square assemblies known as horizontally monotone assemblies. This situation guarantees a connected partition, demonstrating that certain structural properties of assemblies can lead to efficient solutions. The uncovering of NP-completeness and a specialized algorithm presents a rich duality between intractability in the general problem and efficient solutions in constrained scenarios.

Detailed Analysis

The results extend the theoretical understanding of assembly planning problems by providing rigorous proofs and detailed algorithmic strategies. The realization that the connected partition problem can be decisively constrained by geometric and combinatorial factors lays foundation for further exploration, not just in 2D but in potential extensions to 3D spaces. While the NP-completeness result firmly establishes the problem’s complexity, the constructive methods and algorithmic design showcase pathways to practical solutions in computer-aided design systems—specifically when immediate feedback in design-for-assembly processes is necessary.

When considering practical implementations, particularly in robotics and automated manufacturing environments, the implications of these algorithms become paramount. The ability to assess whether partitions can be achieved without collisions, ensuring subassembly connectivity, directly ties into the feasibility of automated systems to efficiently manufacture complex assemblies with minimal human intervention.

Future Directions and Implications

The findings invite further inquiry into broader assembly planning scenarios, such as those involving multi-step partitions or variably constrained environments. As the algorithm is effective for planar polygons and links to programmable matter through unit-grid squares, there remains substantial room for developing heuristic or approximation techniques applicable to more generalized or 3D problems.

The non-trivial NP-completeness and FPT results in this context not only impact assembly planning but also encourage exploration of connectivity constraints in other geometric and combinatorial settings. Subsequent exploration and technique refinement in these areas could lead to improved design automation tools that can handle a wider array of constraints seamlessly, promoting efficiency in both design and production phases.

In conclusion, the research by Agarwal and colleagues signifies an important development in the domain of assembly planning and geometric computational complexity. By resolving a long-standing question in computational geometry and providing a tractable approach for practical instances, this work constructs a robust framework around which further improvements in both assembly planning algorithms and their applications can be designed.