A Simple and Near-Optimal Algorithm for Directed Expander Decompositions

Konsep Inti

The author presents a new algorithm for computing expander decompositions in directed graphs with near-optimal time complexity, simplifying previous approaches while maintaining efficiency.

Abstrak

The content introduces a novel algorithm for directed expander decompositions in graphs. It improves upon existing methods by achieving near-optimal runtimes and simplifying the process. The algorithm is based on a push-pull-relabel flow framework that generalizes classic algorithms and enhances efficiency. By maintaining valid states and utilizing witness graphs, the algorithm ensures the graph remains an expander even under edge deletions. The approach involves reducing the problem to maintaining out-expanders, allowing for efficient maintenance of expander decompositions.

Statistik

In this article, we use ˜O(·) notation to suppress factors logarithmic in m, i.e. O(m logc m) = ˜O(m) for every constant c > 0.
Given a parameter φ ≤ c/ log2 m for a fixed constant c > 0, and a directed m-edge graph G undergoing a sequence of edge deletions, there is a randomized data structure that constructs and maintains a (O(log6 m), Ω(φ/ log8 m))-expander decomposition (X, Er) of G.
If the sequence of edge deletions is of length at most m/2, then our algorithm further has the property that it updates X such that it is refining over time meaning that at any current time, X is a refinement of its earlier versions.

Kutipan

"Our result improves over previous algorithms [BGS20, HKGW23] that only obtained algorithms optimal up to subpolynomial factors."
"In this work, we study the problem of computing and maintaining expander decompositions in directed graphs."
"Our new techniques are much simpler and more accessible than previous work, besides also being much faster."

Wawasan Utama Disaring Dari

A Simple and Near-Optimal Algorithm for Directed Expander Decompositions

by Aurelio L. S... pada arxiv.org 03-08-2024

https://arxiv.org/pdf/2403.04542.pdf

Pertanyaan yang Lebih Dalam

How can the push-pull-relabel framework be applied to other graph-related problems

The push-pull-relabel framework can be applied to various graph-related problems that involve flow optimization, such as network flow algorithms, max-flow min-cut problems, and circulation problems. By adapting the push-pull-relabel technique, one can efficiently handle dynamic changes in graphs while maintaining optimal or near-optimal solutions. This framework is particularly useful in scenarios where there are constraints on flow capacities and the need to balance flows across different vertices.

What are potential drawbacks or limitations of using witness graphs in maintaining expander decompositions

While witness graphs play a crucial role in maintaining expander decompositions by providing evidence of graph expansion properties, they also come with certain drawbacks and limitations. One limitation is the computational overhead involved in constructing and updating these witness graphs as part of the algorithm. Additionally, ensuring that the witness graphs accurately represent the expansion properties of the original graph can be challenging and may require additional resources for verification.

How does the concept of out-expanders simplify the process of maintaining expander decompositions

Out-expanders simplify the process of maintaining expander decompositions by focusing on out-going edges from a given set of vertices rather than considering all edges within a graph. This approach reduces complexity by narrowing down the scope of analysis to specific directional relationships between vertices. By concentrating on out-expansion properties, algorithms can more efficiently identify sparse cuts and maintain overall graph connectivity without needing to consider bidirectional interactions extensively. This simplification streamlines computations and updates related to expander decompositions in directed graphs.