Solving Connected Subgraph Problems in Wildlife Conservation

Abstract

Abstract. We investigate mathematical formulations and solution techniques for a variant of the Connected Subgraph Problem. Given a connected graph with costs and profits associated with the nodes, the goal is to find a connected subgraph that contains a subset of distinguished vertices. In this work we focus on the budget-constrained version, where we maximize the total profit of the nodes in the subgraph subject to a budget constraint on the total cost. We propose several mixed-integer formulations for enforcing the subgraph connectivity requirement, which plays a key role in the combinatorial structure of the problem. We show that a new formulation based on subtour elimination constraints is more effective at capturing the combinatorial structure of the problem, providing significant advantages over the previously considered encoding which was based on a single commodity flow. We test our formulations on synthetic instances as well as on real-world instances of an important problem in environmental conservation concerning the design of wildlife corridors. Our encoding results in a much tighter LP relaxation, and more importantly, it results in finding better integer feasible solutions as well as much better upper bounds on the objective (often proving optimality or within less than 1% of optimality), both when considering the synthetic instances as well as the real-world wildlife corridor instances.

Networks, uncertainty, applications and a crusade for optimality

“…design [see, e.g., Conrad et al, 2012, Dilkina andGomes, 2010]. Here, the nodes correspond to land parcels, their weights are associated with the habitat suitability, and node costs are associated with land value.…”

“…[Moss and Rabani, 2007] have proposed an O(log n) approximation algorithm for the B-RMWCS with non-negative node-weights, where n is the number of nodes in the graph. For more details on the problems related on the RMWCS, see e.g., the literature review given in [Dilkina and Gomes, 2010].…”

“…Previously studied mixed integer programming (MIP) formulations for the (B-)RMWCS use arc and possibly flow variables to model the problem [see Dilkina and Gomes, 2010]. In this work we propose three new MIP models for the (B-)RMWCS derived in the natural space of node variables.…”

“…A budgeted version arises in conservation planning, where the task is to select land parcels for conservation to ensure species viability, also called corridor design [Dilkina and Gomes, 2010]. Here, the nodes of the graph do not only have node weights associated with the habitat suitability but also some costs, and the task is to design wildlife corridors that maximize the suitability with a given limited budget.…”

Learning to Run Heuristics in Tree Search

“…Applications of Mixed Integer Programming (MIP) have thus spanned domains as diverse as aircraft routing [Barnhart et al, 1998], wildlife conservation [Dilkina and Gomes, 2010], sports scheduling [Nemhauser and Trick, 1998], dose distribution [Lee et al, 1999] and kidney exchange [Abraham et al, 2007], to mention a few. As such, * Corresponding author, bdilkina@cc.gatech.edu improving the performance of MIP solvers can have a dramatic impact across various domains.…”