Placing Green Bridges Optimally for Robust Habitat Reconnection

Abstract

We study the problem of robustly reconnecting habitats via the placement of green bridges at minimum total cost. Habitats are fragmented into patches and we seek to reconnect each habitat such that it remains connected even if any of its patches becomes unavailable. Formally, we are given an undirected graph with edge costs, a set of fixed green bridges represented as a subset of the graph's edges, a set of habitats represented as vertex subsets, and some budget. We decide whether there exists a subset of the graph's edges containing all fixed green bridges such that, for each habitat, the induced subgraph on the solution edges is 2-vertex-connected, and the total cost does not exceed the budget. We also study the 2-edge-connectivity variant, modeling the case where any single reconnecting green bridge may fail. We analyze the computational complexity of these problems, focusing on the boundary between NP-hardness and polynomial-time solvability when the maximum habitat size and maximum vertex degree are bounded by constants. We prove that for each constant maximum habitat size of at least four there exists a small constant maximum degree for which the problems are NP-hard, and complement this with polynomial-time algorithms yielding partial dichotomies for bounded habitat size and degree.

Figures (22)

• Figure 1: Overview of our results for (left) RGBP and (right) the edge-connectivity version of RGBP. Green corresponds to polynomial-time solvability, red to $\operatorname{NP}$-hardness even with unit costs and no forced edges. ($^\dagger$: cf. \ref{['rem:H14D4:NPhard']})
• Figure 2: Illustrative example to \ref{['def:HG']}. (Left) A graph $G$, a habitat set $\mathcal{H}$ with a blue, orange, magenta, and green size-4 habitat (contained vertices are encircled), and the set $F_{\rm in}$ of forced edges consisting of the thick/red edges. (Right) The corresponding basic habitat graph $\mathcal{G}_{G,\mathcal{H},F_{\rm in}}$ with two components each being a path of length one.
• Figure 3: Illustration to \ref{['constr:H4D7:NPhard']}. (a) $V_i$ and habitat $H_i$ (cyan). (b) $V_{i,j}$ with its neighbors in $V_i$ and $V_j$, as well as the habitats $H_{i,j}$ (orange), $H_{i,j}'$ (cyan), and $H_{i,j}"$ (magenta). (c) $V_i$, $V_j$, and $V_{i,j}$, as well as the habitats $H_{i,j}^1$ (cyan), $H_{i,j}^2$ (orange), $H_{i,j}^3$ (orange), $H_{i,j}^4$ (teal), and $H_{i,j}^5$ (blue). (d) An excerpt from $G'$ around $V_i$, where green edges indicate additional edges in a solution. (Every red edge is contained in $E_2'$.)
• Figure 4: Illustration to the three cases (a), (b), and (c) in the proof of \ref{['lem:H4D5:P:triangle']}.
• Figure 5: Illustration to \ref{['constr:H5D6:NPhard']} (red edges are in every solution). (a) $V_{x_i}$ and the habitats $H_{x_i}^1$ (orange), $H_{x_i}^2$ (magenta), and $H_{x_i}$ (cyan). (b) $W_j$ and the habitats $H_{C_j}^{-1}$ (orange), $H_{C_j}^{-2}$ (magenta), and $H_{C_j}^{-3}$ (cyan) (c) An excerpt around $V_i$ according to $x_i$'s negated appearances with habitats $H_{x_i,C_j}$ (cyan) and $H_{x_i,C_{j'}}$ (orange). Green edges indicate a part of a solution that corresponds to setting $x_i$ to false.

Theorems & Definitions (73)

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