This work concerns an alignment problem that has applications in many geospatial problems such as resource allocation and building reliable disease maps. Here, we introduce the problem of optimally aligning $k$ collections of $m$ spatial supports over $n$ spatial units in a $d$-dimensional Euclidean space. We show that the 1-dimensional case is solvable in time polynomial in $k$, $m$ and $n$. We then show that the 2-dimensional case is NP-hard for 2 collections of 2 supports. Finally, we devise a heuristic for aligning a set of collections in the 2-dimensional case.

翻译：本研究探讨一种在资源分配和构建可靠疾病地图等众多地理空间问题中具有应用价值的对齐问题。本文提出在d维欧几里得空间中，将k组各含m个空间支撑的集合最优对齐到n个空间单元上的问题。我们证明一维情形可在关于k、m和n的多项式时间内求解。进而证明二维情形对于2组各含2个支撑的集合是NP难问题。最后，我们设计了一种用于二维情形下多组集合对齐的启发式算法。