Splitting (complicated) surfaces is hard

Abstract

Let M be an orientable surface without boundary. A cycle on M is splitting if it has no self-intersections and it partitions M into two components, neither homeomorphic to a disk. In other words, splitting cycles are simple, separating, and non-contractible. We prove that finding the shortest splitting cycle on a combinatorial surface is NP-hard but fixed-parameter tractable with respect to the surface genus. Specifically, we describe an algorithm to compute the shortest splitting cycle in g O(g) n 2 log n time.

Minimum cuts and shortest homologous cycles

“…After many intermediate improvements [8,11,27,52], Cabello and Chambers [9] described the fastest algorithm currently known for this problem, which runs in O(g 3 n log n) time. Splitting cycles are non-contractible and separating; finding the shortest such cycle is NP-hard, although there is an O(n log n)-time algorithm for graphs of any fixed genus [12,13]. Colin de Verdière and Erickson [18] prove that the shortest path or cycle in a given homotopy class can be computed in polynomial time, improving earlier results of Colin de Verdière and Lazarus [17,20,21].…”

“…Our algorithm closely resembles the algorithm of Chambers et al [13] for computing a shortest splitting cycle; in fact, our algorithm is somewhat simpler. The first stage of our algorithm cuts the underlying combinatorial surface into a topological disk by a network of shortest paths.…”

“…Cellular graph embeddings are equivalent to the combinatorial surfaces introduced by Colin de Verdière [17] and used by several other authors to formulate optimization problems for surface-embedded graphs [9,10,11,13,18,19,20,21,27,28,29,52]. A combinatorial surface S = (Σ, G) consists of an abstract surface Σ together with a cellularly embedded graph G with positively weighted edges.…”

Computing the Shortest Essential Cycle

“…If the parameters g and b are constants, then for any k, we can compute the shortest k-essential cycle in O(n log n) time using a recent algorithm of Chambers et al [8,Theorem 6.1]. The input to their algorithm is a combinatorial surface Σ and a set of pairs S = { (g 1 , b 1 ), (g 2 , b 2 ), .…”

“…For example, algorithms that repeatedly cut a given surface along short, topologically nontrivial cycles have been used for removing topological noise from graphical models [21], finding short cut graphs for surface parametrization [18], computing shortest paths in a given homotopy class [12], approximating optimal traveling salesman tours in surface-embedded graphs [14], probabilistically embedding high-genus graphs into planar graphs [26,2], drawing abstract graphs in the plane with the fewest possible crossings [28], and testing isomorphism between graphs of fixed genus [27]. These and other applications have motivated a series of algorithms for computing optimal cycles with various topological properties [35,31,18,7,4,29,5,8,6,9].…”