Minimizing the stabbing number of matchings, trees, and triangulations

TL;DR

This paper investigates the computational complexity of minimizing the stabbing number in geometric structures like matchings, trees, and triangulations, proving NP-hardness for several cases and proposing an integer programming approach with promising experimental results.

Contribution

It establishes NP-hardness for several minimum stabbing problems and introduces a cut-based integer programming formulation with effective approximation potential.

Findings

NP-hardness of multiple stabbing number problems

A polynomial-time lower bound via LP relaxations

Successful computational experiments on hundreds of points

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 points. The complexity of these problems has been a long-standing open question; in fact, it is one of the original 30 outstanding open problems in computational geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide is negative for a number of minimum stabbing problems by showing them NP-hard by means of a general proof technique. It implies non-trivial lower bounds 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. We obtain lower bounds (in polynomial time) from the corresponding linear programming relaxations, and show that an optimal fractional solution always contains an edge of at least constant weight. This result constitutes a crucial step towards a constant-factor approximation via an iterated rounding scheme. In computational experiments we demonstrate that our approach allows for actually solving problems with up to several hundred points optimally or near-optimally.