On TwoHanded Planar Assembly Partitioning with Connectivity Constraints
Abstract
The construction used in the hardness result.

Assembly planning 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 that each of the two subassemblies will be connected.
In this paper we study a natural special case of the connectedassemblypartitioning problem: Given a connected set A of unitgrid squares in the plane, find a connected subset S of 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 NPcomplete, settling an open question posed by Wilson et al. (1995) a quarter of a century ago. We complement the hardness result with two positive results. First, we show that it is fixedparameter tractable and give an O(2^{k}n^{2})time algorithm, where n=A and k=S. Second, we describe a special case of this variant where a connected partition can always be found in linear time. Each of the positive results sheds further light on the special geometric structure of the problem at hand.
Links
 Pankaj K. Agarwal, Boris Aronov, Tzvika Geft, and Dan Halperin
On TwoHanded Planar Assembly Partitioning with Connectivity Constraints
In 2021 ACMSIAM Symposium on Discrete Algorithms (SODA 2021)
[SODA] [arXiv]  Videos: [25 minute summary] [Extended talk on algorithmic results] [Extended talk on hardness]
Contacts
Tzvika Geft  
Dan Halperin 