Exact and heuristic solutions for the prize-collecting geometric enclosure problem

In the prize-collecting geometric enclosure problem (PCGEP), a set $S$ of points in the plane is given, each with an associated benefit. The goal is to find a simple polygon $P$ with vertices in $S$ that maximizes the sum of the benefits of the points of $S$ enclosed by $P$ minus the perimeter of $P$ multiplied by a given nonnegative cost. The PCGEP is NP-complete and has applications to land surveying for exploration or preservation of natural resources. In this paper, we develop the first heuristic, called PCGEP-GR, for the PCGEP and revisit a previously proposed integer linear programming (ILP) model to solve it to optimality. We conducted a comprehensive experimental study of that heuristic and an exact algorithm based on the ILP model. We show that a new set of constraints, together with the previous set, is necessary to guarantee the correctness of the ILP model and introduce preprocessing strategies that allow us to prove optimality 40% faster on average. The proposed heuristic is able to reach the optimum in 32% of our benchmark instances and, for those with unknown optima, PCGEP-GR was found better than or at least as good solutions as the ones obtained by the cplex ILP solver in 54% of the cases. Notwithstanding these positive results, the design of effective heuristics for the PCGEP proved to be very challenging, which also led us to obtain a result that provides the theoretical foundation for future advances in the study of this problem.