An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations (1304.2338v2)

Abstract: In this paper, we introduce a new framework for approximately solving flow problems in capacitated, undirected graphs and apply it to provide asymptotically faster algorithms for the maximum $s$-$t$ flow and maximum concurrent multicommodity flow problems. For graphs with $n$ vertices and $m$ edges, it allows us to find an $\epsilon$-approximate maximum $s$-$t$ flow in time $O(m^{1+o(1)}\epsilon^{-2})$, improving on the previous best bound of $\tilde{O}(mn^{1/3} poly(1/\epsilon))$. Applying the same framework in the multicommodity setting solves a maximum concurrent multicommodity flow problem with $k$ commodities in $O(m^{1+o(1)}\epsilon^{-2}k^2)$ time, improving on the existing bound of $\tilde{O}(m^{4/3} poly(k,\epsilon^{-1})$. Our algorithms utilize several new technical tools that we believe may be of independent interest: - We give a non-Euclidean generalization of gradient descent and provide bounds on its performance. Using this, we show how to reduce approximate maximum flow and maximum concurrent flow to the efficient construction of oblivious routings with a low competitive ratio. - We define and provide an efficient construction of a new type of flow sparsifier. In addition to providing the standard properties of a cut sparsifier our construction allows for flows in the sparse graph to be routed (very efficiently) in the original graph with low congestion. - We give the first almost-linear-time construction of an $O(m^{o(1)})$-competitive oblivious routing scheme. No previous such algorithm ran in time better than $\tilde{{\Omega}}(mn)$. We also note that independently Jonah Sherman produced an almost linear time algorithm for maximum flow and we thank him for coordinating submissions.

• The paper introduces an almost-linear-time algorithm that approximates max flow in undirected graphs using innovative non-Euclidean gradient descent and flow sparsifiers.
• The paper extends its approach to solve the multicommodity flow problem in O(m^(1+o(1)) k^2) time, significantly reducing previous computational complexities.
• The paper demonstrates the scalability of its methods, enabling efficient processing of massive networks for applications in telecommunications and logistics.

An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs

This paper presents a significant advancement in the area of flow problems on undirected graphs by introducing a nearly linear-time algorithm for approximating maximum $s$-$t$ flows. Moreover, the authors extend this work to the concurrent multicommodity flow problem, which involves routing multiple commodities through a network simultaneously. By leveraging novel algorithmic frameworks and technical tools, the authors have achieved substantial improvements over previous results in terms of time complexity.

Key Contributions

• Algorithmic Frameworks: The authors propose a new computational strategy for handling flow problems in capacitated networks. This framework enables the approximation of maximum $s$-$t$ flows in time $O(m^{1+o(1)}^{-2})$, a significant improvement compared to the prior best, achieved in $O(mn^{1/3}\mathrm{poly(1/)})$ time.
• Extension to Multicommodity Flows: Utilizing a similar approach, the maximum concurrent multicommodity flow problem for $k$ commodities is solved in $O(m^{1+o(1)}^{-2}k^2)$ time, improving considerably over the previously best-known time complexity of $O(m^{4/3}\mathrm{poly}(k,^{-1}))$.
• Technical Innovations: The authors introduce several technical concepts:
  • A non-Euclidean generalization of gradient descent is applied, which is competitive for the specific metric defined by the flow problem.
  • Construction of a new type of flow sparsifier, which preserves structural cut properties while allowing effective back-routing of flows into the original network.
  • Development of an almost-linear-time construction for oblivious routing, yielding an $O(m^{o(1)})$-competitive scheme.

These technical contributions have the potential for broader applicability beyond the specific problems addressed in this paper, as they offer fundamental improvements in the structure and execution of flow algorithms.

Numerical Results and Claims

The paper's algorithms are underpinned by rigorous numerical analysis and theoretical proofs. A strong emphasis is placed on the scalability of the methods and their applicability to large graphs with millions of edges and vertices. The reduction in computational complexity to nearly linear time offers practical implementations across various domains involving network flow, including telecommunications, transport, and logistics optimization.

Implications and Future Directions

The presented work not only enhances the theoretical understanding of flow problems but also provides a scalable approach to practical applications that require efficient network analysis. The reduction of time complexity implies that larger and more complex networks can be processed, which aligns well with the expanding scale of modern data networks.

Future research could further refine these methods, potentially extending them to weighted or directed graphs, and exploring their integration with existing combinatorial optimization frameworks. There may be possibilities to adapt these techniques to dynamic or time-varying networks, broadening their applicability even further.

The intersection of efficient graph algorithms, numerical linear algebra, and optimization opens exciting new avenues for deploying these solutions in real-world applications and across computational paradigms, potentially influencing developments in parallel computing and cloud-based network optimization.