The problem of packing smaller objects within a larger object has been of interest since decades. In these problems, in addition to the requirement that the smaller objects must lie completely inside the larger objects, they are expected to not overlap or have minimum overlap with each other. Due to this, the problem of packing turns out to be a non-convex problem, obtaining whose optimal solution is challenging. As such, several heuristic approaches have been used for obtaining sub-optimal solutions in general, and provably optimal solutions for some special instances. In this paper, we propose a novel encoder-decoder architecture consisting of an encoder block, a perturbation block and a decoder block, for packing identical circles within a larger circle. In our approach, the encoder takes the index of a circle to be packed as an input and outputs its center through a normalization layer, the perturbation layer adds controlled perturbations to the center, ensuring that it does not deviate beyond the radius of the smaller circle to be packed, and the decoder takes the perturbed center as input and estimates the index of the intended circle for packing. We parameterize the encoder and decoder by a neural network and optimize it to reduce an error between the decoder's estimated index and the actual index of the circle provided as input to the encoder. The proposed approach can be generalized to pack objects of higher dimensions and different shapes by carefully choosing normalization and perturbation layers. The approach gives a sub-optimal solution and is able to pack smaller objects within a larger object with competitive performance with respect to classical methods.

翻译：将小物体填充到大物体内的问题数十年来一直备受关注。在这类问题中，除了要求小物体必须完全位于大物体内部外，还要求它们彼此不重叠或重叠最小化。由此，填充问题转化为一个非凸问题，其最优解的求解颇具挑战性。为此，人们通常采用多种启发式方法以获得次优解，并在某些特殊情形下得到可证明的最优解。本文提出一种新颖的编码器-解码器架构，包括编码块、扰动块和解码块，用于将相同大小的圆填充至更大的圆内。在该方法中，编码器以待填充圆的索引为输入，通过归一化层输出其圆心；扰动层对圆心施加受控扰动，确保其偏离不超过待填充小圆的半径；解码器以扰动后的圆心为输入，估计待填充目标圆的索引。我们使用神经网络参数化编码器和解码器，并通过优化来减小解码器估计索引与编码器输入圆实际索引之间的误差。通过精心选择归一化和扰动层，该方法可推广至填充更高维度及不同形状的物体。该方法提供次优解，且能以与经典方法相当的性能将小物体填充至大物体内。