TY - GEN

T1 - On two-handed planar assembly partitioning with connectivity constraints

AU - Agarwal, Pankaj K.

AU - Aronov, Boris

AU - Geft, Tzvika

AU - Halperin, Dan

N1 - Funding Information:
P. K. Agarwal has been partially supported by NSF grants IIS-18-14493 and CCF-20-07556.
Publisher Copyright:
Copyright © 2021 by SIAM

PY - 2021

Y1 - 2021

N2 - Assembly planning is a fundamental problem in robotics and automation, which aims to design a sequence of motions that brings the separate constituent parts of a product into their final placement in the product. It is convenient to study assembly planning in reverse order, where the following key problem, assembly partitioning, arises: Given a set of parts in their final placement in a product, partition them into two sets, each regarded as a rigid body, which we call a subassembly, such that these two subassemblies can be moved sufficiently far away from each other, without colliding with one another. The basic assembly planning problem is further complicated by practical consideration such as how to hold the parts in a subassembly together. Therefore, a desired property of a valid assembly partition is for each of the two subassemblies to be connected. In this paper we study a natural special case of the connected-assembly-partitioning problem: Given a connected set A of unit-grid squares in the plane, find a connected subset S ⊂ A such that A \ S is also connected and S can be rigidly translated to infinity along a prescribed direction without colliding with A \ S. We show that even this simple problem is NP-complete, settling an open question posed by Wilson et al. a quarter of a century ago [16]. We complement the hardness result with two positive results. First, we show that the problem is fixed-parameter tractable and present an O(2kn2)-time algorithm, where n = |A| and k = |S|. Second, we describe a special case of this problem where a connected partition can always be found in O(n) time.

AB - Assembly planning is a fundamental problem in robotics and automation, which aims to design a sequence of motions that brings the separate constituent parts of a product into their final placement in the product. It is convenient to study assembly planning in reverse order, where the following key problem, assembly partitioning, arises: Given a set of parts in their final placement in a product, partition them into two sets, each regarded as a rigid body, which we call a subassembly, such that these two subassemblies can be moved sufficiently far away from each other, without colliding with one another. The basic assembly planning problem is further complicated by practical consideration such as how to hold the parts in a subassembly together. Therefore, a desired property of a valid assembly partition is for each of the two subassemblies to be connected. In this paper we study a natural special case of the connected-assembly-partitioning problem: Given a connected set A of unit-grid squares in the plane, find a connected subset S ⊂ A such that A \ S is also connected and S can be rigidly translated to infinity along a prescribed direction without colliding with A \ S. We show that even this simple problem is NP-complete, settling an open question posed by Wilson et al. a quarter of a century ago [16]. We complement the hardness result with two positive results. First, we show that the problem is fixed-parameter tractable and present an O(2kn2)-time algorithm, where n = |A| and k = |S|. Second, we describe a special case of this problem where a connected partition can always be found in O(n) time.

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M3 - Conference contribution

AN - SCOPUS:85105282515

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1740

EP - 1756

BT - ACM-SIAM Symposium on Discrete Algorithms, SODA 2021

A2 - Marx, Daniel

PB - Association for Computing Machinery

T2 - 32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021

Y2 - 10 January 2021 through 13 January 2021

ER -