Maximum Flow and Minimum-Cost Flow in Almost-Linear Time (2203.00671v2)

Abstract: We give an algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with $m$ edges and polynomially bounded integral demands, costs, and capacities in $m^{1+o(1)}$ time. Our algorithm builds the flow through a sequence of $m^{1+o(1)}$ approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized $m^{o(1)}$ time using a new dynamic graph data structure. Our framework extends to algorithms running in $m^{1+o(1)}$ time for computing flows that minimize general edge-separable convex functions to high accuracy. This gives almost-linear time algorithms for several problems including entropy-regularized optimal transport, matrix scaling, $p$-norm flows, and $p$-norm isotonic regression on arbitrary directed acyclic graphs.

• The paper introduces an almost-linear time algorithm that computes exact maximum and minimum-cost flows on directed graphs via iterative approximate minimum-ratio cycles.
• It leverages a novel interior point method framework and low-stretch trees to achieve performance close to theoretical optimality for large-scale network flow problems.
• The approach extends to complex applications such as entropy-regularized optimal transport and matrix scaling, highlighting its versatility in real-world optimization challenges.

An Algorithm for Maximum Flow and Minimum-Cost Flow in Almost-Linear Time

The paper presents a novel algorithm that computes exact maximum flows and minimum-cost flows on directed graphs in almost-linear time, specifically $m^{1+o(1)}$, with applications to various flow-related problems. This work builds on theoretical advancements in solving flow problems more efficiently by leveraging dynamic graph data structures and a new interior point method (IPM) framework. Notably, the algorithm is applicable to directed graphs with polynomially bounded integral demands, costs, and capacities, providing a performance guarantee close to the best-known theoretical bounds.

Key Contributions

1. Algorithmic Framework: The proposed algorithm employs a new iterative framework that constructs flows through a sequence of $m^{1+o(1)}$ approximate undirected minimum-ratio cycles. The cycles are computed and processed efficiently using a dynamic graph data structure over the sequence of iterations. This innovative use of minimum-ratio cycles, as opposed to traditional augmenting paths or blocking flows, marks a significant departure from conventional approaches for solving flow problems.
2. Efficiency and Applicability: The algorithm's running time is almost-linear, matching the optimal time up to subpolynomial factors for several flow-related problems. It scales efficiently with the number of edges $m$ in the graph, which is particularly beneficial in large-scale applications where the number of edges is substantial. Furthermore, the framework extends to algorithms that solve more complex flow problems, such as flows minimizing general edge-separable convex functions, with high accuracy.
3. Data Structure Innovations: A dynamic graph data structure is developed to handle the changing nature of the graph as the algorithm progresses. This structure maintains approximate solutions to minimum-ratio cycle subproblems, crucial for the iterative process of building the flow. The data structure's design ensures that the updates required after processing each cycle are efficient, enabling the almost-linear time complexity.
4. Extensions and Generalizations: Beyond maximum flow and minimum-cost flow, the algorithm extends to address problems like entropy-regularized optimal transport, matrix scaling, and $p$-norm flows on arbitrary directed acyclic graphs. This versatility demonstrates the robustness of the algorithmic framework and highlights its potential impact across various domains needing efficient flow solutions.

Technical Contributions

1. Practical and Theoretical Implications: The development of an efficient algorithm for min-cost flow with only a logarithmic dependence on the capacity range $U$ has notable implications for practice, enabling faster computations in applications such as network optimization, logistics, and resource allocation. The theoretical underpinnings also provide insights into the structure of flow problems, offering new directions for both algorithmic design and complexity analysis.
2. Role of Interior Point Methods: The paper revisits the powerful Interior Point Method techniques, adjusting them to work efficiently within the unique constraints and structures of flow problems. This adaptation reflects a deeper understanding of how continuous optimization techniques can be integrated with discrete problem solving.
3. Low-Stretch Trees and Embeddings: A significant aspect of the algorithm lies in the innovative use of low-stretch trees and embeddings, which facilitate efficient exploration of flow paths. This technique reduces the complexity of cycle computations and improves overall algorithm efficiency.
4. Stability and Robustness: The algorithm demonstrates stability relative to the perturbations in graph structure, ensuring reliable performance even under dynamically changing graph conditions. The robust handling of unstable cycle subproblems showcases the careful consideration of stability in the design of iterative optimization algorithms.

Future Directions

The research opens several future avenues in both theoretical and applied optimization. The reliance on dynamic data structures suggests further refinements could yield even more efficient iterations, aiding applications spanning from real-time traffic navigation to large-scale logistics planning. Exploring the algorithm's adaptability to other problem domains requiring optimization under constraints could reveal new potential for this almost-linear time framework.

Overall, this paper contributes both a significant advancement in flow algorithm efficiency and a framework fertile for further exploration, continuing to bridge gaps between theoretical optimization and practical problem-solving. The insights gained here could inspire subsequent work aimed at further reducing the complexity of network flow problems and extending applicability across various domains.