Maximum Flow by Augmenting Paths in $n^{2+o(1)}$ Time
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arXiv
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| Format: | Preprint |
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2024
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| author | Bernstein, Aaron Blikstad, Joakim Saranurak, Thatchaphol Tu, Ta-Wei |
| author_facet | Bernstein, Aaron Blikstad, Joakim Saranurak, Thatchaphol Tu, Ta-Wei |
| contents | We present a combinatorial algorithm for computing exact maximum flows in directed graphs with $n$ vertices and edge capacities from $\{1,\dots,U\}$ in $n^{2+o(1)}\log U$ time, which is almost optimal in dense graphs. Our algorithm is a novel implementation of the classical augmenting-path framework; we list augmenting paths more efficiently using a new variant of the push-relabel algorithm that uses additional edge weights to guide the algorithm, and we derive the edge weights by constructing a directed expander hierarchy.
Even in unit-capacity graphs, this breaks the long-standing $O(m\cdot\min\{\sqrt{m},n^{2/3}\})$ time bound of the previous combinatorial algorithms by Karzanov (1973) and Even and Tarjan (1975) when the graph has $m=ω(n^{4/3})$ edges. Notably, our approach does not rely on continuous optimization nor heavy dynamic graph data structures, both of which are crucial in the recent developments that led to the almost-linear time algorithm by Chen et al. (FOCS 2022). Our running time also matches the $n^{2+o(1)}$ time bound of the independent combinatorial algorithm by Chuzhoy and Khanna (STOC 2024) for computing the maximum bipartite matching, a special case of maximum flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03648 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximum Flow by Augmenting Paths in $n^{2+o(1)}$ Time Bernstein, Aaron Blikstad, Joakim Saranurak, Thatchaphol Tu, Ta-Wei Data Structures and Algorithms We present a combinatorial algorithm for computing exact maximum flows in directed graphs with $n$ vertices and edge capacities from $\{1,\dots,U\}$ in $n^{2+o(1)}\log U$ time, which is almost optimal in dense graphs. Our algorithm is a novel implementation of the classical augmenting-path framework; we list augmenting paths more efficiently using a new variant of the push-relabel algorithm that uses additional edge weights to guide the algorithm, and we derive the edge weights by constructing a directed expander hierarchy. Even in unit-capacity graphs, this breaks the long-standing $O(m\cdot\min\{\sqrt{m},n^{2/3}\})$ time bound of the previous combinatorial algorithms by Karzanov (1973) and Even and Tarjan (1975) when the graph has $m=ω(n^{4/3})$ edges. Notably, our approach does not rely on continuous optimization nor heavy dynamic graph data structures, both of which are crucial in the recent developments that led to the almost-linear time algorithm by Chen et al. (FOCS 2022). Our running time also matches the $n^{2+o(1)}$ time bound of the independent combinatorial algorithm by Chuzhoy and Khanna (STOC 2024) for computing the maximum bipartite matching, a special case of maximum flow. |
| title | Maximum Flow by Augmenting Paths in $n^{2+o(1)}$ Time |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2406.03648 |