Faster Cut Sparsification of Weighted Graphs

Area of Science:

• Theoretical Computer Science
• Graph Algorithms
• Computational Complexity

Background:

• Cut sparsifiers are reweighted subgraphs that approximate original graph cut weights.
• Existing algorithms for computing cut sparsifiers have limitations in running time, especially for unbounded weights.
• Efficient cut sparsification is crucial for developing fast approximate min-cut algorithms.

Purpose of the Study:

• To develop a more efficient algorithm for computing cut sparsifiers for weighted graphs.
• To achieve improved time complexity for cut sparsification, particularly for unbounded integer weights.
• To establish new state-of-the-art results for approximate min-cut algorithms.

Main Methods:

• The algorithm computes cut sparsifiers of size O(n log(n) / ε^2).
• It achieves a running time of O(m * min(α(n)log(m/n), log(n))), improving upon previous O(m log^2(n)) methods.
• The approach adapts existing algorithms for unweighted graphs by utilizing partial maximum spanning forest (MSF) packings.

Main Results:

• The proposed algorithm computes cut sparsifiers in significantly improved time complexity.
• This yields the best known results for cut sparsification on graphs with unbounded integer weights.
• It also leads to the fastest known approximate min-cut algorithms for both polynomial and unbounded weighted graphs.

Conclusions:

• The study presents a novel and efficient algorithm for computing cut sparsifiers.
• The adaptation of MSF packings is key to achieving these improved results.
• This work advances the field of graph algorithms and approximation techniques for cut problems.