Deterministic enumeration of all minimum <i>k</i>-cut-sets in hypergraphs for fixed <i>k</i>

Beideman, Calvin; Chandrasekaran, Karthekeyan; Wang, Weihang

doi:10.1137/1.9781611977073.88

Cited by 2 publications

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“…Our algorithms for both Enum-MinMax-Hypergraph-k-Partition and Enum-Hypergraphk-Cut are based on a structural theorem that allows for efficient recovery of optimum k-cut-sets via minimum (s, t)-terminal cuts (see Theorem 1.4). Our structural theorem builds on structural theorems that have appeared in previous works on Minmax-Hypergraph-k-Partition and Hypergraph-k-Cut [7,11,12]. Our structural theorem may appear to be natural/incremental in comparison to ones that appeared in previous works, but formalizing the theorem and proving it is a significant part of our contribution.…”

Section: Resultsmentioning

confidence: 91%

“…A subsequent deterministic algorithm was designed to solve Hypergraph-k-Cut in time n O(k) p by Chandrasekaran and Chekuri [11]. Chandrasekaran and Chekuri's techniques were extended to design the first deterministic polynomial-time algorithm to solve Enum-Hypergraph-k-Cut in [7]. The algorithm for Enum-Hypergraph-k-Cut given in [7] runs in time n O(k 2 ) p. We note that this run-time has a quadratic dependence on k in the exponent of n although the number of min-k-cut-sets has only linear dependence on k in the exponent of n (since it is O(n 2k−2 )).…”

Section: Resultsmentioning

confidence: 99%

“…Chandrasekaran and Chekuri's techniques were extended to design the first deterministic polynomial-time algorithm to solve Enum-Hypergraph-k-Cut in [7]. The algorithm for Enum-Hypergraph-k-Cut given in [7] runs in time n O(k 2 ) p. We note that this run-time has a quadratic dependence on k in the exponent of n although the number of min-k-cut-sets has only linear dependence on k in the exponent of n (since it is O(n 2k−2 )). So, an open question that remained from [7] is whether one can obtain an n O(k) p-time deterministic algorithm for Enum-Hypergraph-k-Cut.…”

Section: Resultsmentioning

confidence: 99%

“…The powerful modeling capability of hyperedges has been useful in a variety of modern applications, which in turn, has led to a resurgence in the study of hypergraphs with recent works focusing on min-cuts, cut-sparsifiers, spectral-sparsifiers, etc. [6,7,11,14,18,19,23,24,35,42,56]. Our work adds to this rich and emerging theory of hypergraphs.…”

Section: Introductionmentioning

confidence: 84%

“…Queyranne claimed, in 1999, a polynomial-time algorithm for Minsum-SymSubmodk-Partition for every fixed k [52], however the claim was retracted subsequently (see [27]). The complexity status of submodular k-partitioning problems (for fixed k ≥ 4) are open, so recent works have focused on hypergraph k-partitioning problems as a stepping stone towards submodular k-partitioning [7,11,12,27,50,60,61]. Our work contributes to this stepping stone by advancing the state of the art in hypergraph k-partitioning problems.…”

Section: Introductionmentioning

confidence: 99%

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Counting and enumerating optimum cut sets for hypergraph $k$-partitioning problems for fixed $k$

Beideman¹,

Chandrasekaran²,

Wang³

2022

Preprint

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We consider the problem of enumerating optimal solutions for two hypergraph k-partitioning problems-namely, Hypergraph-k-Cut and Minmax-Hypergraph-k-Partition. The input in hypergraph k-partitioning problems is a hypergraph G = (V, E) with positive hyperedge costs along with a fixed positive integer k. The goal is to find a partition of V into k non-empty parts (V 1 , V 2 , . . . , V k )-known as a k-partition-so as to minimize an objective of interest.1. If the objective of interest is the maximum cut value of the parts, then the problem is known as Minmax-Hypergraph-k-Partition. A subset of hyperedges is a minmax-kcut-set if it is the subset of hyperedges crossing an optimum k-partition for Minmax-Hypergraph-k-Partition. 2. If the objective of interest is the total cost of hyperedges crossing the k-partition, then the problem is known as Hypergraph-k-Cut. A subset of hyperedges is a min-k-cut-set if it is the subset of hyperedges crossing an optimum k-partition for Hypergraph-k-Cut.

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Section: Resultsmentioning

confidence: 91%

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Counting and enumerating optimum cut sets for hypergraph $k$-partitioning problems for fixed $k$

Beideman¹,

Chandrasekaran²,

Wang³

2022

Preprint

Self Cite

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

Fox

Panigrahi

Zhang

2023

ACM Trans. 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 $\widetilde{O}\left(mn^{2k-2}\right) $ 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 $\widetilde{O}\left(n^{\max \lbrace r,2k-2\rbrace }\right) $ 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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Deterministic enumeration of all minimum k-cut-sets in hypergraphs for fixed k

Cited by 2 publications

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Counting and enumerating optimum cut sets for hypergraph $k$-partitioning problems for fixed $k$

Counting and enumerating optimum cut sets for hypergraph $k$-partitioning problems for fixed $k$

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

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