Integer programming approaches for minimum stabbing problems

TL;DR: This is the first computational study ever reported in which these three stabbing problems are considered and where provably optimal solutions are given, and integer programming (ip) formulations for these three problems, that allowed them to solve them to optimality through ip branch-and-bound (b&b) or branch- and-cut ( b&c) algorithms.

Abstract: The problem of finding structures with minimum stabbing number has received considerable attention from researchers. Particularly, [10] study the minimum stabbing number of perfect matchings (mspm), spanning trees (msst) and triangulations (mstr) associated to set of points in the plane. The complexity of the mstr remains open whilst the other two are known to be 𝓝𝓟-hard. This paper presents integer programming (ip) formulations for these three problems, that allowed us to solve them to optimality through ip branch-and-bound (b&b) or branch-and-cut (b&c) algorithms. Moreover, these models are the basis for the development of Lagrangian heuristics. Computational tests were conducted with instances taken from the literature where the performance of the Lagrangian heuristics were compared with that of the exact b&b and b&c algorithms. The results reveal that the Lagrangian heuristics yield solutions with minute, and often null, duality gaps for instances with several hundreds of points in small computation times. To our knowledge, this is the first computational study ever reported in which these three stabbing problems are considered and where provably optimal solutions are given.