[Paper Review] Minimum Cut of Directed Planar Graphs in O(nloglogn) Time

Shay Mozes, Cyril Nikolaev+2 more|arXiv (Cornell University)|Dec 7, 2015

Complexity and Algorithms in Graphs8 references3 citations

TL;DR

This paper presents an O(n log log n) time algorithm for computing the minimum cut in a weighted directed planar graph, leveraging a novel adaptation of the CFN recursive separator framework and exploiting a surprising duality: when the minimum st-cut is also the global minimum cut, a shortest-path-based min-st-cut algorithm works in the directed setting. The key contribution is the first sub-quadratic improvement for directed planar min cut, making it faster than the current O(n log n) min st-cut algorithm.

ABSTRACT

We give an $O(n \log \log n)$ time algorithm for computing the minimum cut (or equivalently, the shortest cycle) of a weighted directed planar graph. This improves the previous fastest $O(n\log^3 n)$ solution. Interestingly, while in undirected planar graphs both min-cut and min $st$-cut have $O(n \log \log n)$ solutions, in directed planar graphs our result makes min-cut faster than min $st$-cut, which currently requires $O(n \log n)$.

Motivation & Objective

• To close the performance gap between min cut and min st-cut in directed planar graphs, where min st-cut was previously faster.
• To develop an efficient algorithm for computing the minimum cut in weighted directed planar graphs.
• To exploit a structural duality in directed planar graphs where the shortest-path-based min-st-cut algorithm works when the min-st-cut is also the global minimum cut.
• To achieve an O(n log log n) time complexity, matching the best known bound for undirected planar min cut.
• To generalize the approach to graphs embedded on surfaces of bounded genus, providing near-optimal algorithms for shortest cycle detection.

Proposed method

• Adapt the CFN recursive separator framework from undirected to directed planar graphs by computing both min st-cut and min ts-cut at each recursive step.
• Leverage a key insight: the shortest-path-based min-st-cut algorithm works correctly in the directed case if the min-st-cut is also the global minimum cut.
• Use a divide-and-conquer strategy based on shortest path separators in the dual graph, maintaining crossing parity information across holes via embedded paths.
• Construct a recursion graph that mirrors the structure of the dual distance-decomposition graph (DDG), preserving embedding and crossing parity for efficient partitioning.
• Employ dynamic data structures to maintain shortest path information and support efficient queries during recursion.
• Generalize the algorithm to graphs on surfaces of bounded genus g using a greedy system of loops or a planarizing set, combining planar and non-planar cycle detection.

Experimental results

Research questions

• RQ1 Can the shortest-path-based min-st-cut algorithm be adapted to work in directed planar graphs under specific structural conditions?
• RQ2 Why does the current O(n log n) min st-cut algorithm remain faster than min cut in directed planar graphs, and can this gap be closed?
• RQ3 Is it possible to achieve O(n log log n) time for minimum cut in directed planar graphs using recursive separator techniques?
• RQ4 Can the duality between minimum cuts and shortest cycles in planar graphs be exploited to solve the min cut problem more efficiently in the directed case?
• RQ5 How can algorithms for shortest cycle detection be extended to graphs embedded on surfaces of bounded genus?

Key findings

• The paper achieves an O(n log log n) time algorithm for computing the minimum cut in a weighted directed planar graph, significantly improving over the previous O(n log³n) bound.
• The algorithm exploits a previously overlooked structural property: when the minimum st-cut is also the global minimum cut, the shortest-path-based min-st-cut algorithm works correctly in the directed setting.
• The recursive framework from the undirected CFN algorithm is adapted to the directed case by computing both min st-cut and min ts-cut at each recursion level.
• The algorithm maintains a recursion graph that preserves the embedding and crossing parity of paths across holes, enabling linear-time partitioning of subgraphs.
• For graphs embedded on surfaces of genus g, the paper presents two algorithms: one running in O(g²n log n) time with high probability, and another in O(gn log²n) time in the worst case.
• The results establish that min cut in directed planar graphs can now be solved faster than min st-cut, reversing the previous performance hierarchy.