Subcubic Algorithms for Gomory–Hu Tree in Unweighted Graphs

Amir Abboud; Robert Krauthgamer; Ohad Trabelsi

doi:10.1145/3406325.3451073

Subcubic Algorithms for Gomory–Hu Tree in Unweighted Graphs

Amir Abboud
, Robert Krauthgamer
, Ohad Trabelsi

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

Every undirected graph G has a (weighted) cut-equivalent tree T, commonly named after Gomory and Hu who discovered it in 1961. Both T and G have the same node set, and for every node pair s,t, the minimum (s,t)-cut in T is also an exact minimum (s,t)-cut in G. We give the first subcubic-time algorithm that constructs such a tree for a simple graph G (unweighted with no parallel edges). Its time complexity is O(n2.5), for n=|V(G)|; previously, only O(n3) was known, except for restricted cases like sparse graphs. Consequently, we obtain the first algorithm for All-Pairs Max-Flow in simple graphs that breaks the cubic-time barrier. Gomory and Hu compute this tree using n-1 queries to (single-pair) Max-Flow; the new algorithm can be viewed as a fine-grained reduction to O(?n) Max-Flow computations on n-node graphs.

Original language	English
Title of host publication	STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
Editors	Samir Khuller, Virginia Vassilevska Williams
Publisher	Association for Computing Machinery
Pages	1725-1737
Number of pages	13
ISBN (Electronic)	9781450380539
DOIs	https://doi.org/10.1145/3406325.3451073
State	Published - 15 Jun 2021
Externally published	Yes
Event	53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021 - Virtual, Online, Italy Duration: 21 Jun 2021 → 25 Jun 2021

Publication series

Name	Proceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)	0737-8017

Conference

Conference	53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021
Country/Territory	Italy
City	Virtual, Online
Period	21/06/21 → 25/06/21

Bibliographical note

Publisher Copyright:
© 2021 ACM.

Keywords

Gomory-Hu trees
maximum flow
minimum cut

ASJC Scopus subject areas

Software

Access to Document

10.1145/3406325.3451073

Cite this

Abboud, A., Krauthgamer, R., & Trabelsi, O. (2021). Subcubic Algorithms for Gomory–Hu Tree in Unweighted Graphs. In S. Khuller, & V. V. Williams (Eds.), STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (pp. 1725-1737). (Proceedings of the Annual ACM Symposium on Theory of Computing). Association for Computing Machinery. https://doi.org/10.1145/3406325.3451073

Abboud, Amir ; Krauthgamer, Robert ; Trabelsi, Ohad. / Subcubic Algorithms for Gomory–Hu Tree in Unweighted Graphs. STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. editor / Samir Khuller ; Virginia Vassilevska Williams. Association for Computing Machinery, 2021. pp. 1725-1737 (Proceedings of the Annual ACM Symposium on Theory of Computing).

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title = "Subcubic Algorithms for Gomory–Hu Tree in Unweighted Graphs",

abstract = "Every undirected graph G has a (weighted) cut-equivalent tree T, commonly named after Gomory and Hu who discovered it in 1961. Both T and G have the same node set, and for every node pair s,t, the minimum (s,t)-cut in T is also an exact minimum (s,t)-cut in G. We give the first subcubic-time algorithm that constructs such a tree for a simple graph G (unweighted with no parallel edges). Its time complexity is O(n2.5), for n=|V(G)|; previously, only O(n3) was known, except for restricted cases like sparse graphs. Consequently, we obtain the first algorithm for All-Pairs Max-Flow in simple graphs that breaks the cubic-time barrier. Gomory and Hu compute this tree using n-1 queries to (single-pair) Max-Flow; the new algorithm can be viewed as a fine-grained reduction to O(?n) Max-Flow computations on n-node graphs.",

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author = "Amir Abboud and Robert Krauthgamer and Ohad Trabelsi",

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Abboud, A, Krauthgamer, R & Trabelsi, O 2021, Subcubic Algorithms for Gomory–Hu Tree in Unweighted Graphs. in S Khuller & VV Williams (eds), STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. Proceedings of the Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, pp. 1725-1737, 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, Virtual, Online, Italy, 21/06/21. https://doi.org/10.1145/3406325.3451073

Subcubic Algorithms for Gomory–Hu Tree in Unweighted Graphs. / Abboud, Amir; Krauthgamer, Robert; Trabelsi, Ohad.
STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. ed. / Samir Khuller; Virginia Vassilevska Williams. Association for Computing Machinery, 2021. p. 1725-1737 (Proceedings of the Annual ACM Symposium on Theory of Computing).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Abboud, Amir

AU - Krauthgamer, Robert

AU - Trabelsi, Ohad

PY - 2021/6/15

Y1 - 2021/6/15

N2 - Every undirected graph G has a (weighted) cut-equivalent tree T, commonly named after Gomory and Hu who discovered it in 1961. Both T and G have the same node set, and for every node pair s,t, the minimum (s,t)-cut in T is also an exact minimum (s,t)-cut in G. We give the first subcubic-time algorithm that constructs such a tree for a simple graph G (unweighted with no parallel edges). Its time complexity is O(n2.5), for n=|V(G)|; previously, only O(n3) was known, except for restricted cases like sparse graphs. Consequently, we obtain the first algorithm for All-Pairs Max-Flow in simple graphs that breaks the cubic-time barrier. Gomory and Hu compute this tree using n-1 queries to (single-pair) Max-Flow; the new algorithm can be viewed as a fine-grained reduction to O(?n) Max-Flow computations on n-node graphs.

AB - Every undirected graph G has a (weighted) cut-equivalent tree T, commonly named after Gomory and Hu who discovered it in 1961. Both T and G have the same node set, and for every node pair s,t, the minimum (s,t)-cut in T is also an exact minimum (s,t)-cut in G. We give the first subcubic-time algorithm that constructs such a tree for a simple graph G (unweighted with no parallel edges). Its time complexity is O(n2.5), for n=|V(G)|; previously, only O(n3) was known, except for restricted cases like sparse graphs. Consequently, we obtain the first algorithm for All-Pairs Max-Flow in simple graphs that breaks the cubic-time barrier. Gomory and Hu compute this tree using n-1 queries to (single-pair) Max-Flow; the new algorithm can be viewed as a fine-grained reduction to O(?n) Max-Flow computations on n-node graphs.

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ER -

Abboud A, Krauthgamer R, Trabelsi O. Subcubic Algorithms for Gomory–Hu Tree in Unweighted Graphs. In Khuller S, Williams VV, editors, STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. Association for Computing Machinery. 2021. p. 1725-1737. (Proceedings of the Annual ACM Symposium on Theory of Computing). doi: 10.1145/3406325.3451073

Subcubic Algorithms for Gomory–Hu Tree in Unweighted Graphs

Abstract

Publication series

Conference

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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Cite this