# The Circle Packing Problem

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The single container pAs a well-known NP-hard problem, the circle packing problem is to pack n given items with no overlap into a two-dimensional container such that the container size is minimized. The packing problems aim to find the most possible dense packing patterns for the given items in a single container . The container can be square, circle, semicircle, polygon, cubs and rectangle, and the items can be rectangles, circles or irregular. As an important class of optimization problems, the packing problems haveis wide applications ly used in the industry and science fieldﬁelds, and it has a wide application field, such as applied mathematics, material manufacturing, material cutting, logistics, architecture layout, wireless communication,…show more content…
So several researchers generally have recourse to heuristic methods. For the packing problem, the two-dimensional container can be square, circle, semicircle, polygon, cubs and rectangle and the items can be rectangles, circles or irregular. However, as an NP-hard problem, there is no exact algorithm to obtain optimality in polynomial time unless N = NP, and researchers have resorted to heuristics or approximation methods.
As a well-known NP-hard problem, the circle packing problem We consider two-dimensional (2D) circle packing problem (CPP) where items are circles. is to pack n given items with no overlap into a two-dimensional container such that the container size is minimized.
In this paper the items to be packed are circles whose radii are not uniform. TheCPP problem, especially the circles packing problem (CPP) that the two-dimensional container is square, has been well an article of researchstudied by a wide range of different researchers. In 2009 Mhand and Rym [54] flexedreview the most relevant literature on efficient models and methods for packing circular items in different types of containers. Packing problem CPP is classified into two categories basing on whether the circle items are , equal [85,6,7] circles or unequal [138,9,10] circles, and there are other variants by considering additional constraints, such as . Besides, there is an important extension, the circle packing problem with equilibrium constraints (CPPEC) [611,