A faster algorithm for the k-forest problem: breaking the Ok(n3/2) complexity barrier

2024-09-30

arXiv.org

AI Summary

This paper studies the problem of maximizing the number of edges in a union of $k$ edge-disjoint forests in a directed graph, subject to the constraint that each vertex has in-degree at most $k$. It also addresses two closely related problems: maximizing the number of edges in a $k$-forest union in undirected graphs, and completing a directed graph to be strongly $k$-connected. The work breaks the long-standing $O_k(n^{3/2})$ time barrier by introducing three novel subproblem frameworks—directed $k$-forests (with in-degree constraints), $k$-quasi-forests, and top-level cluster computation—and integrating combinatorial optimization, hierarchical contraction, dynamic trees, and multi-stage reductions, with amortized analysis eliminating logarithmic overheads. The resulting algorithm runs in $O(k^3 min{kn, m} log^2 n + k cdot ext{MAXFLOW}(m,m) log n)$ time, significantly improving upon all prior algorithms for sparse graphs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The $k$-forest problem asks to find $k$ forests in a graph $G$ maximizing the number of edges in their union. We show how to solve this problem in $O(k^3 min{kn, m} log^2 n + k cdot{ m MAXFLOW}(m, m) log n)$ time, breaking the $O_k(n^{3/2})$ complexity barrier of previously known approaches. Our algorithm relies on three subroutines: the directed $k$-forest problem with bounded indegree condition, the $k$-pseudoforest problem, and the top clump computation.

Problem

• Develop faster algorithms for packing edge-disjoint forests in directed graphs
• Improve time complexity for k-forest problem in undirected graphs
• Enhance directed edge-connectivity augmentation with more efficient solutions

Innovation

• Min-max characterization for packing edge-disjoint forests
• Almost linear time algorithm for fixed parameter k
• Improved complexities for connectivity augmentation problems