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An alignment problem

Emma L. McDaniel, Armin R. Mikler, Chetan Tiwari, Murray Patterson

TL;DR

It is shown that the 1-dimensional case is solvable in time polynomial in time polynomial in k, m, and n, and that the 2-dimensional case is NP-hard for 2 collections of 2 supports.

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.

An alignment problem

TL;DR

It is shown that the 1-dimensional case is solvable in time polynomial in time polynomial in k, m, and n, and that the 2-dimensional case is NP-hard for 2 collections of 2 supports.

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 collections of spatial supports over spatial units in a -dimensional Euclidean space. We show that the 1-dimensional case is solvable in time polynomial in , and . 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.

Paper Structure

This paper contains 8 sections, 1 theorem, 13 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Tractability results
  3. Hardness results
  4. Claim.
  5. A heuristic for the 2-dimensional case
  6. Aligning a pair of collections in 2 dimensions
  7. The Alignment Problem in 2 dimensions
  8. Conclusion

Key Result

Theorem 1

Problem prob:alignment in $d$-dimensions is NP-hard for $k,m,d \ge 2$.

Figures (4)

  • Figure 1: Collections (a) $\mathcal{S} = \{s_1, s_2, s_3, s_4\}$ and (b) $\mathcal{T} = \{t_1, t_2, t_3, t_4\}$ of spatial supports over the set $U = \{u_{1,1}, u_{1,2}, \dots, u_{4,4}\}$ of 16 spatial units. An alignment (c) of $\mathcal{S}$ and $\mathcal{T}$. The red dots mark the units ($u_{2,1}, u_{3,1}, u_{1,2}, u_{2,2}, u_{3,2}, u_{2,3}, u_{2,4}$) on which $\mathcal{S}$ and $\mathcal{T}$ disagree.
  • Figure 2: Four collections of three spatial supports (green, orange, and blue) over the set $U = \{u_1, u_2, \dots, u_9\}$ of 9 spatial units. All disagreements between (the supports of) any pair of collections are contained in the two transparent windows.
  • Figure 3: Base set $U \cup \{a, b\} = \{a, u_1, \dots, u_n, b\}$ of spatial units.
  • Figure 4: The shared units graph $G_x$ (Definition \ref{['def:gx']}) of the instance $\mathcal{S}, \mathcal{T}$ depicted in Figure \ref{['fig:eg']}, where the populations $p_\mathcal{S}(u)$ of each unit $u$ in $s_1, s_2, s_3, s_4$ of $\mathcal{S}$ are 20, 20, 10, 15, respectively, while the populations $p_\mathcal{T}(u)$ of each unit $u$ in $t_1, t_2, t_3, t_4$ of $\mathcal{T}$ are 15, 15, 12, 20, respectively. Edges of zero weight are not shown for easier readability.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Definition 1: Shared units Graph $G_x$
  • Definition 2: Shared units Hypergraph $H_x$