Computing the Geometric Intersection Number of Curves

TL;DR

This paper presents efficient algorithms for computing the geometric intersection number of curves on surfaces, including methods for minimal self-intersections, homotopic curve construction, and simple curve detection, with implications for classical problems.

Contribution

The paper introduces the first exact polynomial-time algorithms for computing the geometric intersection number and related problems on surfaces, improving upon previous complexity bounds.

Findings

Algorithms run in O(n+ℓ^2) time for intersection number computation.

Constructs homotopic curves realizing minimal intersections in O(n+ℓ^4) time.

Decides if a curve is homotopic to a simple curve in O(n+ℓ log ℓ) time.

Abstract

The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve $c$ represented by a closed walk of length at most $\ell$ on a combinatorial surface of complexity $n$ we describe simple algorithms to (1) compute the geometric intersection number of $c$ in $O(n+ \ell^2)$ time, (2) construct a curve homotopic to $c$ that realizes this geometric intersection number in $O(n+\ell^4)$ time, (3) decide if the geometric intersection number of $c$ is zero, i.e. if $c$ is homotopic to a simple curve, in $O(n+\ell\log\ell)$ time. The algorithms for (2) and (3) are restricted to orientable surfaces, but the algorithm for (1) is also valid on non-orientable surfaces. To our knowledge, no exact complexity analysis had yet appeared on those problems. An optimistic analysis of the complexity of the published algorithms for problems (1) and (3) gives at best a $O(n+g^2\ell^2)$ time complexity on a genus $g$ surface without boundary. No polynomial time algorithm was known for problem (2) for surfaces without boundary. Interestingly, our solution to problem (3) provides a quasi-linear algorithm to a problem raised by Poincar\'e more than a century ago. Finally, we note that our algorithm for problem (1) extends to computing the geometric intersection number of two curves of length at most $\ell$ in $O(n+ \ell^2)$ time.