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.