# Minimum Cuts and Shortest Cycles in Directed Planar Graphs via Noncrossing Shortest Paths

## TL;DR

This paper presents faster algorithms for finding minimum cuts and shortest cycles in directed planar graphs by leveraging duality and noncrossing shortest paths, improving previous time bounds to nearly linearithmic.

## Contribution

It introduces an $O(n ext{log}n ext{log} ext{log}n)$-time algorithm for both problems using a divide-and-conquer approach based on noncrossing shortest paths.

## Key findings

- Achieved $O(n ext{log}n ext{log} ext{log}n)$ time complexity for minimum cut and shortest cycle problems.
- Extended noncrossing shortest path algorithms from undirected to directed planar graphs.
- Provided a unified approach exploiting duality in planar graphs to improve algorithmic efficiency.

## Abstract

Let $G$ be an $n$-node simple directed planar graph with nonnegative edge weights. We study the fundamental problems of computing (1) a global cut of $G$ with minimum weight and (2) a~cycle of $G$ with minimum weight. The best previously known algorithm for the former problem, running in $O(n\log^3 n)$ time, can be obtained from the algorithm of \Lacki, Nussbaum, Sankowski, and Wulff-Nilsen for single-source all-sinks maximum flows. The best previously known result for the latter problem is the $O(n\log^3 n)$-time algorithm of Wulff-Nilsen. By exploiting duality between the two problems in planar graphs, we solve both problems in $O(n\log n\log\log n)$ time via a divide-and-conquer algorithm that finds a shortest non-degenerate cycle. The kernel of our result is an $O(n\log\log n)$-time algorithm for computing noncrossing shortest paths among nodes well ordered on a common face of a directed plane graph, which is extended from the algorithm of Italiano, Nussbaum, Sankowski, and Wulff-Nilsen for an undirected plane graph.