Placing Green Bridges Optimally, with a Multivariate Analysis

TL;DR

This work models the placement of wildlife crossings as a graph-structured optimization problem across three connectivity variants, focusing on how data quality (via the reach parameter $d$) influences tractability. It delivers an $O(mn+rnd)$-time ${$rd$}-approximation for the $d$-Reach GBP and provides a comprehensive complexity map: $\,\mathrm{NP}$-hardness for $d\ge1$ in many cases, fixed-parameter tractability for $d\le2$ with kernels, and $\\,\mathrm{W[1]}$-hardness for $d\ge3$ when parameterizing by $k$ and $r$. The results illuminate the practical feasibility of different models under varying data quality and habitat configurations and establish kernelization bounds, including a polynomial kernel for planar graphs in the $d=2$ case and linear-time solvability for the two-habitat $d=1$ scenario. The insights have bearing on how to structure data collection and which connectivity requirements to enforce in real-world conservation planning.

Abstract

We study the problem of placing wildlife crossings, such as green bridges, over human-made obstacles to challenge habitat fragmentation. The main task herein is, given a graph describing habitats or routes of wildlife animals and possibilities of building green bridges, to find a low-cost placement of green bridges that connects the habitats. We develop different problem models for this task and study them from a computational complexity and parameterized algorithmics perspective.

Figures (9)

• Figure 1: A diagram of interconnections between the problems (for the definition of Connect GBP see \ref{['prob:congbp']}). An edge from problem $A$ to problem $B$ means that any solution to $A$ is also a solution to $B$. Problems with $d$ omitted from the problem name require that there is a solution for some value of $d$.
• Figure 2: Illustration to \ref{['constr:1rgbp-planar']} for [1] on series-parallel (and thus planar) graphs. In this example, there are e.g. $F_p\supseteq\{1,i,j\}$ and $F_q\supseteq\{i,j,n\}$. In case of a yes-instance, the red-colored edges are in every solution (\ref{['obs:1rgbp-planar']}).
• Figure 3: Illustration to \ref{['constr:1rgbpNPhconsthab-A', 'constr:1rgbpNPhconsthab-B']}. Part (a) shows an exemplary directed graph which is a yes-instance for DHP. Applying \ref{['constr:1rgbpNPhconsthab-A']} on (a) yields (b). Applying \ref{['constr:1rgbpNPhconsthab-B']} on (b) yields the instance whose graph is depicted in (c) and two habitats of which are depicted in (d) and (e). Vertices marked yellow in (d) are contained in the habitat $X_{\rm out}$. Vertices marked red in (e) are contained in the habitat $Y_{\rm out}$. The graph induced by $Y_{\rm out}$ contains the red edges.
• Figure 4: Illustration for [2] with (a) $r=2$ and $\Delta=4$ ($k'=m+(n-1)+k$) and (b) $r=1$ ($k'=m+k$).
• Figure 5: Illustration for the construction in the proof of \ref{['prop:rgbp:nopk']} for [2] with $r=1$. In this example, $U=\{u_1,\dots,u_n\}$ and we have $\{u_1,u_i,u_j,u_n\}= F\in\mathcal{F}$.

Theorems & Definitions (51)

• Theorem 1
• Lemma 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Lemma 6
• proof
• Proposition 7