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This paper formulates two variants of packing problems in which the set of items is known but the arrival order is unknown. The goal is to certify that the items can be packed in a given container, and/or to optimize the size or cost of a container so that that the items are guaranteed to be packable, regardless of arrival order. The Non-Deterministically Ordered Packing (NDOP) variant asks to generate a certificate that a packing plan exists for every ordering of items. QuasiOnline Packing (QOP) asks to generate a partially-observable packing policy that chooses the location of each item as their arrival order is revealed one-by-one, with all of the remaining items certified to be packable regardless of their future arrival order. Theoretical analysis demonstrates that even the simple subproblem of verifying feasibility of a packing policy is NPcomplete. Despite this worst-case complexity, practical solvers for both NDOP and QOP are developed. Multiple extensions to the basic nondeterministic problem are presented, including packing with a fixed-capacity buffer and packing with equivalent objects. Experiments demonstrate that these algorithms can be applied to packing irregular 3D shapes with stability and manipulator loading constraints. Note to Practitioners: Abstract—Automatic packing of objects into containers, such as a shipping box, has applications in warehouse automation for e-commerce applications and lessthan-truckload (LTL) shipping. In many scenarios, a container needs to be pre-selected for a set of objects before they arrive, and the order of the objects cannot be controlled. This paper studies the theoretical foundations of packing problems in which the set of items is known but the arrival order is unknown. The algorithms presented in this paper can certify whether a container can hold a set of objects in the case where the arrival order is revealed at once, one-by-one, and where a limited set of objects can be put aside in a buffer before packing. These algorithms can be used to choose optimal containers and packing strategies, both for robot and human packers.

Robot Packing with Known Items and Nondeterministic Arrival Order