Random Contractions and Sampling for Hypergraph and Hedge Connectivity

Ghaffari, Mohsen; Karger, David R.; Panigrahi, Debmalya

doi:10.1137/1.9781611974782.71

Cited by 20 publications

(34 citation statements)

References 9 publications

(19 reference statements)

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“…Finally, better algorithms are known for small values of k ∈ [2, 6] [NI92, HO94, BG97, Kar00, NI00, NKI00, Lev00]. The Karger-Stein algorithm was recently extended to Hypergraph k-Cut [GKP17,CXY18], which also gave a bound on the number of minimum k-cuts. For the minimum cut (k = 2), the number of and the structure of approximate min-cuts also have been studied [HW96,BG08].…”

Section: Other Related Workmentioning

confidence: 99%

The number of minimum k -cuts: improving the Karger-Stein bound

Gupta

Lee

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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Given an edge-weighted graph, how many minimum k-cuts can it have? This is a fundamental question in the intersection of algorithms, extremal combinatorics, and graph theory. It is particularly interesting in that the best known bounds are algorithmic: they stem from algorithms that compute the minimum k-cut.In 1994, Karger and Stein obtained a randomized contraction algorithm that finds a minimum k-cut in O(n (2−o(1))k ) time. It can also enumerate all such k-cuts in the same running time, establishing a corresponding extremal bound of O(n (2−o(1))k ). Since then, the algorithmic side of the minimum k-cut problem has seen much progress, leading to a deterministic algorithm based on a tree packing result of Thorup, which enumerates all minimum k-cuts in the same asymptotic running time, and gives an alternate proof of the O(n (2−o(1))k ) bound. However, beating the Karger-Stein bound, even for computing a single minimum k-cut, has remained out of reach.In this paper, we give an algorithm to enumerate all minimum k-cuts in O(n (1.981+o(1))k ) time, breaking the algorithmic and extremal barriers for enumerating minimum k-cuts. To obtain our result, we combine ideas from both the Karger-Stein and Thorup results, and draw a novel connection between minimum k-cut and extremal set theory. In particular, we give and use tighter bounds on the size of set systems with bounded dual VC-dimension, which may be of independent interest.

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Section: Other Related Workmentioning

confidence: 99%

The number of minimum k -cuts: improving the Karger-Stein bound

Gupta

Lee

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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“…Unfortunately, even if hyperedges are unweighted, contracting a hyperedge chosen uniformly at random may destroy a min-cut with fairly high probability when the hyperedges have variable sizes. Indeed, Kogan and Krauthgamer [15] show that this natural strategy finds a min-cut with probability exponentially small in the rank r. Recently, Ghaffari, Karger, and Panigrahi [5] showed how to achieve Karger's Ω 1/n 2 probability of success by biasing the selection of hyperedges away from those of large size. Chandrasekaran, Xu, and Yu [1] refined this strategy and generalized it to work for k-cuts as well.…”

Section: Randomized Contractions In Hypergraphsmentioning

confidence: 99%

“…In this paper, we design a randomized branching framework for finding minimum cut and minimum k-cut in hypergraphs, leading to improvements in randomized algorithms for these problems over recent results of Ghaffari et al [5] and Chandrasekaran et al [1]. Our algorithms can be thought of as the natural analog in hypergraphs of the celebrated recursive contraction algorithm of Karger and Stein [12] for finding minimum cut and minimum k-cut in graphs.…”

Section: Introductionmentioning

confidence: 99%

Minimum Cut and Minimum k-Cut in Hypergraphs via Branching Contractions

Fox¹,

Panigrahi²,

Zhang³

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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On hypergraphs with m hyperedges and n vertices, where p denotes the total size of the hyperedges, we provide the following results:• We give an algorithm that runs in O mn 2k−2 time for finding a minimum k-cut in hypergraphs of arbitrary rank. This algorithm betters the previous best running time for the minimum k-cut problem, for k > 2.• We give an algorithm that runs in O n max{r,2k−2} time for finding a minimum k-cut in hypergraphs of constant rank r. This algorithm betters the previous best running times for both the minimum cut and minimum k-cut problems for dense hypergraphs.Both of our algorithms are Monte Carlo, i.e., they return a minimum k-cut (or minimum cut) with high probability. These algorithms are obtained as instantiations of a generic branching randomized contraction technique on hypergraphs, which extends the celebrated work of Karger and Stein on recursive contractions in graphs.Our techniques and results also extend to the problems of minimum hedge-cut and minimum hedge-k-cut on hedgegraphs, which generalize hypergraphs.

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“…It is a relatively simple consequence of fundamental decomposition theorems of Cunningham and Edmonds [14], Fujishige [22], and Cunningham [13] on submodular functions from the early 1980s. Recently Ghaffari et al also proved this fact through a random contraction algorithm [26].…”

Section: All Mincuts and The Hypercactus Representationmentioning

confidence: 88%

“…This fact is not explicitly stated in [12,13]. As we already mentioned, the preceding corollary is also derived via a random contraction algorithm in [26].…”

Section: For Each Mincut S Of Hmentioning

confidence: 96%

Minimum Cuts and Sparsification in Hypergraphs

Chekuri¹,

Xu²

2018

SIAM J. Comput.

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We study algorithmic and structural aspects of connectivity in hypergraphs. Given a hypergraph H = (V, E) with n = |V |, m = |E| and p = 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 + n 2 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 (trimmed) subhypergraph H with sum of degrees O(kn) that preserves all edge-connectivities up to k (a k-sparse certificate). This generalizes the corresponding result of Nagamochi and Ibaraki from graphs to hypergraphs. Using this sparsification we obtain an O(p + λn 2) time algorithm for computing a global minimum cut of H where λ is the minimum cut value. • We show that a hypercactus representation of all the global minimum cuts of a capacitated hypergraph can be computed in O(np + n 2 log n) time and O(p) space matching the asymptotic time to find a single minimum cut. • We obtain a (2 +)-approximation to the global minimum cut of a capacitated hypergraph in O(1 (p log n + n log 2 n)) time, and for uncapacitated hypergraphs in O(p/) time. We achieve this by generalizing Matula's algorithm for graphs to hypergraphs. • We describe an algorithm to compute approximate strengths of all the edges of a hypergraph in O(p log 2 n log p) time. This gives a near linear time algorithm for finding a (1 +)-cut sparsifier based on the work of Kogan and Krauthgamer. As a byproduct we obtain faster algorithms for various cut and flow problems in hypergraphs of small rank. 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 orderings for hypergraphs and these yield different insights.

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Random Contractions and Sampling for Hypergraph and Hedge Connectivity

Cited by 20 publications

References 9 publications

The number of minimum k -cuts: improving the Karger-Stein bound

The number of minimum k -cuts: improving the Karger-Stein bound

Minimum Cut and Minimum k-Cut in Hypergraphs via Branching Contractions

Minimum Cuts and Sparsification in Hypergraphs

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