An alignment problem

TL;DR

This paper investigates an alignment problem in geospatial contexts, demonstrating polynomial solvability in 1D, NP-hardness in 2D, and proposing a heuristic for the 2D case.

Contribution

It introduces the formal alignment problem for multiple spatial collections, analyzes its computational complexity, and provides a heuristic solution for the 2D case.

Findings

1D case is solvable in polynomial time.

2D case is NP-hard for specific instances.

A heuristic algorithm for 2D alignment is proposed.

Abstract

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.