research-article

Computing minimum cuts in hypergraphs

Authors:

Chandra Chekuri,

Chao XuAuthors Info & Claims

SODA '17: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

Pages 1085 - 1100

Published: 16 January 2017 Publication History

Abstract

We study algorithmic and structural aspects of connectivity in hypergraphs. Given a hypergraph H = (V, E) with n = |V|, m = |E| and p = Σ_e∈E |e| the fastest known algorithm to compute a global minimum cut in H runs in O(np) time for the uncapacitated case, and in O(np + n² log n) time for the capacitated case. We show the following new results.

• Given an uncapacitated hypergraph H and an integer k we describe an algorithm that runs in O(p) time to find a subhypergraph H' with sum of degrees O(kn) that preserves all edge-connectivities up to k (a k-sparsifier). This generalizes the corresponding result of Nagamochi and Ibaraki from graphs to hypergraphs. Using this sparsification we obtain an O(p + λn²) time algorithm for computing a global minimum cut of H where λ is the minimum cut value.

• We generalize Matula's argument for graphs to hypergraphs and obtain a (2 + ε)-approximation to the global minimum cut in a capacitated hypergraph in O(1/ε (p log n + n log² n)) time, and in in O(p/ε) time for uncapacitated hypergraphs.

• We show that a hypercactus representation of all the global minimum cuts of a capacitated hypergraph can be computed in O(np + n² log n) time and O(p) space.

Our results build upon properties of vertex orderings that were inspired by the maximum adjacency ordering for graphs due to Nagamochi and Ibaraki. Unlike graphs we observe that there are several different orderings for hypergraphs which yield different insights.

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Cited By

Fox KPanigrahi DZhang FChan T(2019)Minimum cut and minimum k-Cut in hypergraphs via branching contractionsProceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3310435.3310489(881-896)Online publication date: 6-Jan-2019
https://dl.acm.org/doi/10.5555/3310435.3310489
Li PMilenkovic O(2018)Revisiting decomposable submodular function minimization with incidence relationsProceedings of the 32nd International Conference on Neural Information Processing Systems10.5555/3327144.3327151(2242-2252)Online publication date: 3-Dec-2018
https://dl.acm.org/doi/10.5555/3327144.3327151
Chandrasekaran KXu CYu XCzumaj A(2018)Hypergraph k-cut in randomized polynomial timeProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175399(1426-1438)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175399

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cover image ACM Conferences

SODA '17: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

January 2017

2756 pages

Program Chair:
Philip N. Klein
Brown University

Sponsors

SIAM Activity Group on Discrete Mathematics
SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 16 January 2017

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SODA '17

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SODA '17: Symposium on Discrete Algorithms

January 16 - 19, 2017

Barcelona, Spain

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Overall Acceptance Rate 411 of 1,322 submissions, 31%

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Cited By

Fox KPanigrahi DZhang FChan T(2019)Minimum cut and minimum k-Cut in hypergraphs via branching contractionsProceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3310435.3310489(881-896)Online publication date: 6-Jan-2019
https://dl.acm.org/doi/10.5555/3310435.3310489
Li PMilenkovic O(2018)Revisiting decomposable submodular function minimization with incidence relationsProceedings of the 32nd International Conference on Neural Information Processing Systems10.5555/3327144.3327151(2242-2252)Online publication date: 3-Dec-2018
https://dl.acm.org/doi/10.5555/3327144.3327151
Chandrasekaran KXu CYu XCzumaj A(2018)Hypergraph k-cut in randomized polynomial timeProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175399(1426-1438)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175399

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