Integer programming approaches for minimum stabbing problems

¹ Universidade Federal de Sergipe, Departamento de Computação, Cidade Universitária Prof. Aloísio Campos, 49100-000 São Cristóvão ( SE), Brazil.
bpiva@ufs.br
² Universidade Estadual de Campinas, Instituto de Computação, Av. Albert Einstein 1251, 13083-852 Campinas ( SP), Brazil.
cid@ic.unicamp.br
³ Universidade Federal Fluminense, Instituto de Computação, R. Passo da Pátria 156, Bloco E, 24210-240 Niterói ( RJ), Brazil.
yuri@ic.uff.br,luidi@ic.uff.br

Received: 11 September 2013
Accepted: 28 November 2013

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.