[Paper Review] Minimum Cut in $O(m\log^2 n)$ Time

Paweł Gawrychowski, Shay Mozes+1 more|arXiv (Cornell University)|Nov 4, 2019

Computational Geometry and Mesh Generation35 references8 citations

TL;DR

This paper presents a deterministic O(m log n) time algorithm for finding the minimum cut that 2-respects a given spanning tree in an undirected weighted graph, significantly improving upon Karger's O(m log² n) bound. By introducing a novel decomposition combining heavy path decomposition with biased divide-and-conquer and a compact topologically induced subtree representation, the authors achieve efficient batch processing of link-cut tree operations, leading to an overall O(m log² n) time randomized algorithm for the minimum cut problem in weighted graphs.

ABSTRACT

We give a randomized algorithm that finds a minimum cut in an undirected weighted $m$-edge $n$-vertex graph $G$ with high probability in $O(m \log^2 n)$ time. This is the first improvement to Karger's celebrated $O(m \log^3 n)$ time algorithm from 1996. Our main technical contribution is a deterministic $O(m \log n)$ time algorithm that, given a spanning tree $T$ of $G$, finds a minimum cut of $G$ that 2-respects (cuts two edges of) $T$.

Motivation & Objective

• To close a 20+ year open problem in minimum cut algorithms: whether Karger's O(m log² n) time bound for the 2-respecting problem could be improved by reducing link-cut tree operations to constant amortized time.
• To design a faster, deterministic algorithm for finding the minimum cut that 2-respects a given spanning tree T in an undirected weighted graph.
• To enable a faster randomized minimum cut algorithm by improving the bottleneck subproblem in Karger's framework.
• To develop a new tree decomposition and compact representation (topologically induced subtrees) that allow efficient batch processing of dynamic tree operations.
• To achieve a log log n speedup in the overall minimum cut algorithm by combining the new 2-respecting algorithm with Karger’s sampling and parameter tuning technique.

Proposed method

• Introduce a new tree decomposition that combines heavy path decomposition with biased divide-and-conquer to partition the tree into manageable fragments.
• Define topologically induced subtrees as a compact representation that preserves the cost structure of cuts across subtrees, enabling efficient batch updates.
• Reframe the 2-respecting problem as a collection of bipartite problems, where the goal is to find the optimal pair of edges (e, e') across two subtrees.
• Process link-cut tree operations in batches using the topologically induced subtree representation, avoiding per-operation O(log n) cost by exploiting static structure and known operation sequences.
• Use a recursive divide-and-conquer strategy on the tree fragments, propagating costs of non-tree edges in bottom-up traversals to maintain edge costs in the compact representation.
• Apply a modified sampling technique from Karger’s framework with tuned parameters (ϵ = 1/(4 log^γ n)) to reduce the number of trees in the packing, enabling a log log n speedup in the overall algorithm.

Experimental results

Research questions

• RQ1 Can the O(m log² n) time bound for the 2-respecting minimum cut problem be improved by exploiting the static nature of the tree and the known sequence of operations?
• RQ2 Is it possible to design a deterministic algorithm for the 2-respecting problem that runs in O(m log n) time, thereby improving the overall minimum cut algorithm to O(m log² n)?
• RQ3 Can a new tree decomposition and compact representation (topologically induced subtrees) be used to batch-process dynamic tree operations and reduce the amortized cost per operation?
• RQ4 Does combining the new 2-respecting algorithm with Karger’s sampling and parameter tuning technique yield a further log log n speedup in the overall minimum cut algorithm?
• RQ5 Can the framework be adapted to unweighted graphs to improve existing randomized algorithms, particularly those relying on Karger’s O(m log³ n) subroutine?

Key findings

• The paper presents a deterministic O(m log n) time algorithm for finding the minimum cut that 2-respects a given spanning tree T, which is the first improvement over Karger’s O(m log² n) algorithm in over 25 years.
• The algorithm achieves this by introducing a novel tree decomposition and a compact representation called topologically induced subtrees, which allows efficient batch processing of dynamic tree operations.
• The authors resolve Karger’s open problem by showing that link-cut tree operations can be batched and processed in amortized O(1) time per operation, leveraging the static structure and known operation sequence.
• By plugging this new 2-respecting algorithm into Karger’s framework, the overall randomized minimum cut algorithm runs in O(m log² n) time, improving upon the previous O(m log³ n) bound.
• A further log log n speedup is achieved by combining the new algorithm with Karger’s sampling technique, resulting in an O(m log² n / log log n + n log^{3+δ} n) time algorithm for any δ > 0.
• The improved algorithm also leads to an O(min{m + n log² n, m log n}) time randomized algorithm for unweighted graphs, improving upon prior results that relied on Karger’s O(m log³ n) subroutine.