Optimally cutting a surface into a disk

TL;DR

This paper studies the problem of cutting a polyhedral surface into a disk with minimal cuts, proving NP-hardness, providing an exact algorithm with exponential complexity, and offering a greedy approximation algorithm.

Contribution

It establishes NP-hardness for the problem, introduces an exact algorithm with specific complexity, and proposes a greedy approximation method.

Findings

NP-hardness of the minimal cut problem on surfaces

An exact algorithm with runtime n^{O(g+k)}

A greedy O(log^2 g)-approximation algorithm

Abstract

We consider the problem of cutting a set of edges on a polyhedral manifold surface, possibly with boundary, to obtain a single topological disk, minimizing either the total number of cut edges or their total length. We show that this problem is NP-hard, even for manifolds without boundary and for punctured spheres. We also describe an algorithm with running time n^{O(g+k)}, where n is the combinatorial complexity, g is the genus, and k is the number of boundary components of the input surface. Finally, we describe a greedy algorithm that outputs a O(log^2 g)-approximation of the minimum cut graph in O(g^2 n log n) time.