Tightening Curves on Surfaces Monotonically with Applications

TL;DR

This paper establishes a polynomial bound on the number of monotonic homotopy moves needed to tighten curves on surfaces, introducing new algorithms for curve minimality and graph reduction that improve upon prior exponential bounds.

Contribution

It provides the first polynomial bound for tightening curves on surfaces and introduces efficient algorithms for minimal multicurve positioning and surface graph reduction.

Findings

Polynomial bound on homotopy moves for curves on surfaces

Efficient polynomial-time algorithms for multicurve minimal position

Unified approach to surface graph reduction using electrical transformations

Abstract

We prove the first polynomial bound on the number of monotonic homotopy moves required to tighten a collection of closed curves on any compact orientable surface, where the number of crossings in the curve is not allowed to increase at any time during the process. The best known upper bound before was exponential, which can be obtained by combining the algorithm of de Graaf and Schrijver [J. Comb. Theory Ser. B, 1997] together with an exponential upper bound on the number of possible surface maps. To obtain the new upper bound we apply tools from hyperbolic geometry, as well as operations in graph drawing algorithms---the cluster and pipe expansions---to the study of curves on surfaces. As corollaries, we present two efficient algorithms for curves and graphs on surfaces. First, we provide a polynomial-time algorithm to convert any given multicurve on a surface into minimal position. Such an algorithm only existed for single closed curves, and it is known that previous techniques do not generalize to the multicurve case. Second, we provide a polynomial-time algorithm to reduce any $k$-terminal plane graph (and more generally, surface graph) using degree-1 reductions, series-parallel reductions, and $\Delta Y$-transformations for arbitrary integer $k$. Previous algorithms only existed in the planar setting when $k \le 4$, and all of them rely on extensive case-by-case analysis based on different values of $k$. Our algorithm makes use of the connection between electrical transformations and homotopy moves, and thus solves the problem in a unified fashion.