Minimizing the Stabbing Number of Matchings, Trees, and Triangulations

Abstract

The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of vertices. The complexity of finding a spanning tree of minimum stabbing number is one of the original 30 questions on “The Open Problems Project” list of outstanding problems in computational geometry by Demaine, Mitchell, and O’Rourke. We show $\mathcal{N}\mathcal{P}$-hardness of stabbing problems by means of a general proof technique. For matchings, this also implies a nontrivial lower bound on the approximability. On the positive side, we propose a cut-based integer programming formulation for minimizing the stabbing number of matchings and spanning trees. From the corresponding linear programming relaxation we obtain polynomial-time lower bounds and show that there always is an optimal fractional solution that contains an edge of at least constant weight. We conjecture that the resulting iterated rounding scheme constitutes a constant-factor approximation algorithm.