Placing Green Bridges Optimally, with Close-Range Habitats in Sparse Graphs

TL;DR

The paper investigates a diameter-2 connectivity problem for reconnecting fragmented habitats via edge-penalty networks, seeking a cheapest edge set F containing forced edges so that each habitat induces a subgraph of diameter at most 2. It develops a nuanced, multi-regime analysis: polynomial-time algorithms for subcubic graphs, for planar graphs with tiny habitats, and for Δ=4 with habitats of size up to four, complemented by NP-hardness results for Δ=5 (even with tiny habitats) and planar instances with modest habitat sizes. Central to the approach are structural tools like the habitat-intersection graph and carefully designed preprocessing rules, enabling reductions to path/cycle DP on components and hierarchical planar reductions. The results delineate sharp tractability borders and provide a foundation for extending diameter-constrained network design to broader graph classes and ecological scenarios with practical implications for wildlife crossing placement.

Abstract

We study a network design problem motivated by the challenge of placing wildlife crossings to reconnect fragmented habitats of animal species, which is among the 17 goals towards sustainable development by the UN: Given a graph, whose vertices represent the fragmented habitat areas and whose edges represent possible green bridge locations (with costs), and the habitable vertex set for each species' habitat, the goal is to find the cheapest set of edges such that each species' habitat is sufficiently connected. We focus on the established variant where a habitat is considered sufficiently connected if it has diameter two in the solution and study its complexity in cases justified by our setting namely small habitat sizes on planar graphs and graphs of small maximum degree $Δ$. We provide efficient algorithms and NP-hardness results for different values of $Δ$ and maximum habitat sizes on general and planar graphs.

Figures (8)

• Figure 1: (a) A road network fragmenting habitats $H_1$, $H_2$, and $H_3$. (b & c) The corresponding graph and a solution (thicker edges) for (b) of minimum cost where the broader road part involves cost 3 (compared to 1 for the others) and the already built green bridge forms a forced edge; (c) of minimum cardinality where edge costs and forced edges are ignored.
• Figure 2: An illustration of the application of \ref{['rr:reducibleH']}, with edge costs written on the edges and $k$ given in the subfigure's label. (a) The habitats are $H'=\{v_\ell,v_r,v_t\}$, $H=\{v_\ell,v_r,v_b\}$, $H_1=\{v_\ell,v_r,v_c\}$, and $H_2=\{v_r,v_b,v_c\}$, where $H$ is the reducible habitat with $H_1$ and $H_2$ being inside $H$. (b) After deleting $H_1$ and $H_2$ and adjusting edge costs and $k$. (c) After dealing with negative edge costs, where edge $\{v_r,v_b\}$ (thick, red) is now forced.
• Figure 3: The zones, where (c)'s $K_3$ must not be contained in (a) or (b).
• Figure 4: Illustration to the cases in the proof of \ref{['thm:maxdeg3']}. In 1), the cases are whether only the red, only the blue, or both paths are present. Dashed edges correspond to subcases.
• Figure 5: Illustration of a graph with three habitats, and the corresponding intersection graph $\partial \mathcal{H}$ by \ref{['def:intersection-graph']}. Note that $H_1$ and $H_3$ are not neighboring in $\partial\mathcal{H}$ as $E(G[H_1 \cap H_3]) \subsetneq E(G[H_1 \cap H_2])$.

Theorems & Definitions (44)

• Example 1
• proof
• Theorem 1
• Definition 2
• Lemma 3
• proof
• proof
• proof : Proof of \ref{['prop:planarH3']}
• Theorem 2
• Definition 5