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- W4313227178 abstract "In 1961, Gomory and Hu showed that the All-Pairs Max-Flow problem of computing the max-flow between all $begin{pmatrix}n2end{pmatrix}$ pairs of vertices in an undirected graph can be solved using only $n-1$ calls to any (single-pair) max-flow algorithm. Even assuming a linear-time max-flow algorithm, this yields a running time of $O(mn)$, which is $O(n^{3})$ when $m=Theta(n^{2})$. While subsequent work has improved this bound for various special graph classes, no subcubic-time algorithm has been obtained in the last 60 years for general graphs. We break this longstanding barrier by giving an $tilde{O}(n^{2})$-time algorithm on general, integer-weighted graphs. Combined with a popular complexity assumption, we establish a counter-intuitive separation: all-pairs max-flows are strictly easier to compute than all-pairs shortest-paths.Our algorithm produces a cut-equivalent tree, known as the Gomory-Hu tree, from which the max-flow value for any pair can be retrieved in near-constant time. For unweighted graphs, we refine our techniques further to produce a Gomory-Hu tree in the time of a poly-logarithmic number of calls to any maxflow algorithm. This shows an equivalence between the all-pairs and single-pair max-flow problems, and is optimal up to polylogarithmic factors. Using the recently announced $m^{1+o(1)}$-time max-flow algorithm (Chen et al., March 2022), our Gomory-Hu tree algorithm for unweighted graphs also runs in $m^{1+o(1)}$-time." @default.
- W4313227178 created "2023-01-06" @default.
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- W4313227178 date "2022-10-01" @default.
- W4313227178 modified "2023-09-27" @default.
- W4313227178 title "Breaking the Cubic Barrier for All-Pairs Max-Flow: Gomory-Hu Tree in Nearly Quadratic Time" @default.
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- W4313227178 doi "https://doi.org/10.1109/focs54457.2022.00088" @default.
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