Shortest Non-trivial Cycles in Directed and Undirected Surface Graphs

Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of non-trivial cycles in G, using different techniques depending on whether or not G is an undirected graph. If G is undirected, then we give an algorithm to compute a shortest non-separating cycle in 2^O(g) n log log n time. Similar algorithms are given to compute a shortest non-contractible or non-null-homologous cycle in 2^O(g+b) n log log n time. Our algorithms for undirected G combine an algorithm of Kutz with known techniques for efficiently enumerating homotopy classes of curves that may be shortest non-trivial cycles. Our main technical contributions in this work arise from assuming G is a directed graph with possibly asymmetric edge weights. For this case, we give an algorithm to compute a shortest non-contractible cycle in G in O((g^3 + g b)n log n) time. In order to achieve this time bound, we use a restriction of the infinite cyclic cover that may be useful in other contexts. We also describe an algorithm to compute a shortest non-null-homologous cycle in G in O((g^2 + g b)n log n) time, extending a known algorithm of Erickson to compute a shortest non-separating cycle. In both the undirected and directed cases, our algorithms improve the best time bounds known for many values of g and b.

💡 Research Summary

The paper addresses the fundamental problem of finding shortest non‑trivial cycles in graphs embedded on orientable surfaces of genus g with b boundary components. A “non‑trivial” cycle may be non‑separating (its removal does not disconnect the surface), non‑contractible (it cannot be continuously shrunk to a point), or non‑null‑homologous (it either separates a pair of boundary cycles or is non‑separating). The authors present a suite of algorithms that dramatically improve the running times for both undirected and directed (asymmetric weight) graphs, achieving near‑linear dependence on the number of vertices n while keeping the dependence on the topological parameters polynomial rather than exponential.

Undirected graphs.
Building on Kutz’s earlier work, the authors observe that one does not need to explore all g‑exponential subsets of the universal cover. By constructing a weighted triangulation of a dualized polygonal schema and enumerating only those homotopy classes that can contain a shortest non‑trivial cycle, they reduce the number of required cover subsets. Consequently, they obtain:

• Shortest non‑separating cycle in 2^{O(g)}·n·log log n time.
• Shortest non‑contractible or shortest non‑null‑homologous cycle in 2^{O(g+b)}·n·log log n time. These results improve upon the previous best of g·2^{O(g)}·n·log log n and match the best known dependence on n for any surface‑embedded problem.

Directed graphs.
Directed graphs pose additional challenges because shortest paths no longer enjoy symmetry or the “single‑crossing” property. The authors first apply the Isolation Lemma to guarantee unique shortest directed paths (perturbing edge weights with infinitesimals). They then use a preprocessing structure due to Cabello, Demaine, and others that, after O(g·n·log n) time and O(n) space, can answer any shortest‑path‑from‑a‑given‑face query in O(log n) time.

1. Shortest non‑null‑homologous cycle.
   Extending Erickson’s algorithm for shortest non‑separating cycles, the authors add O(b) extra copies of a covering space built from shortest paths between boundary components. By running shortest‑path computations in each copy and selecting the minimum, they achieve O((g²+g b)·n·log n) time.

2. Shortest non‑contractible cycle.
   The algorithm distinguishes two cases:

   • If a shortest non‑contractible cycle is non‑separating, Erickson’s O(g²·n·log n) algorithm already finds it.
   • If every shortest non‑contractible cycle is separating, such a cycle lifts to a non‑null‑homologous cycle in a carefully chosen subgraph of the infinite cyclic cover. The infinite cyclic cover is a covering space obtained by repeatedly cutting along a chosen non‑separating loop; the authors restrict attention to a finite “window” of this cover that is sufficient to capture the lifted cycle. By computing shortest paths in this restricted cover and projecting back to the original surface, they obtain a shortest non‑contractible cycle in O((g³+g b)·n·log n) time. This eliminates the exponential dependence on g present in earlier work (e.g., Erickson’s g·2^{O(g)}·n·log n) and yields the first near‑linear‑in‑n algorithm for arbitrary genus.

Technical contributions.

• A novel use of a bounded portion of the infinite cyclic cover, avoiding the full universal cover while still guaranteeing that a shortest separating non‑contractible cycle appears as a non‑null‑homologous cycle in the cover.
• Extension of Erickson’s non‑separating‑cycle framework to handle directed graphs by adding O(b) boundary‑based cover copies.
• Integration of the Cabello‑et‑al. multi‑source‑multi‑sink shortest‑path data structure to achieve O(log n) query time after O(g·n·log n) preprocessing, crucial for keeping the overall dependence on n near‑linear.
• Careful handling of uniqueness of shortest paths via the Isolation Lemma, ensuring that the algorithms’ correctness does not rely on tie‑breaking heuristics.

Impact and future directions.
The presented algorithms close the gap between undirected and directed surface‑graph problems, showing that directed graphs can be handled with comparable efficiency. The techniques—especially the restricted infinite cyclic cover—are likely to be useful for related problems such as minimum s‑t cuts, maximum flows, and homology‑based optimization on surfaces. Moreover, the paper’s modular structure (separate sections for undirected, directed non‑null‑homologous, and directed non‑contractible cycles) makes each result independently applicable, providing a toolbox for researchers tackling a variety of topological graph problems.