Intractability of Assembly Sequencing: Unit Disks in the Plane

We consider the problem of removing a given disk from a collection of unit disks in the plane. At each step, we allow a disk to be removed by a collision-free translation to infinity, and the goal is to access a given disk using as few steps as possible. This Disks problem is a version of a common task in assembly sequencing, namely removing a given part from a fully assembled product. Recently there has been a focus on optimizing assembly sequences over various cost measures, however with very limited algorithmic success. We explain this lack of success, proving strong inapproximability results in this simple geometric setting. Namely, we show that approximating the number of steps required to within a factor of 2 1−γ n for any γ > 0 is quasi-NP-hard. This provides the first inapproximability results for assembly sequencing, realized in a geometric setting. As a stepping stone, we study the approximability of scheduling with and/or precedence constraints. The Disks problem can be formulated as a scheduling problem where the order of removals is to be scheduled. Before scheduling a disk to be removed, a path must be cleared, and so we get precedence constraints on the tasks; however, the form of such constraints differs from traditional scheduling in that there is a choice of which path to clear. We prove our main result by first showing the similar inapproximability of this scheduling problem, and then by showing that a sufficiently hard subproblem can be realized geometrically using unit disks in the plane. Furthermore, our construction is fairly robust, in that is can be placed on a polynomially sized integer grid, it remains valid even when we consider only horizontal and vertical translations, and it also applies to axis-aligned unit squares and higher dimensions. ⋆ Supported by ARO MURI Grant DAAH04-96-1-0007 and by NSF Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. ⋆⋆ Supported by an Alfred P. Sloan Research Fellowship, an IBM Faculty Partnership Award, an ARO MURI Grant DAAH04-96-1-0007, and NSF Young Investigator Award CCR-9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation.