LP Relaxation and Tree Packing for Minimum $k$-cuts

Chekuri, Chandra; Quanrud, Kent; Xu, Chao

doi:10.48550/arxiv.1808.05765

Cited by 5 publications

(7 citation statements)

References 28 publications

(52 reference statements)

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“…This approximation can be extended to a (2 − h/k)-approximation in time n O(h) [XCY11]. Chekuri et al [CQX18] studied the LP relaxation of [NR01] and gave alternate proofs for both approximation and exact algorithms with slightly improved guarantees. A fast (2+ε)-approximation algorithm was also recently given by Quanrud [Qua18].…”

Section: Other Related Workmentioning

confidence: 99%

“…The textbook minimum cut algorithm of Karger and Stein [KS96], based on random edge contractions, can be adapted to solve k-Cut in O(n 2(k−1) ) (randomized) time. The deterministic algorithms side has seen a series of improvements since then [KYN07,Tho08,CQX18]. The fastest algorithm for general edge weights is due to Chekuri et al [CQX18].…”

Section: Introductionmentioning

confidence: 99%

“…The deterministic algorithms side has seen a series of improvements since then [KYN07,Tho08,CQX18]. The fastest algorithm for general edge weights is due to Chekuri et al [CQX18]. It runs in O(mn 2k−3 ) time and is based on a deterministic tree packing result of Thorup [Tho08].…”

Section: Introductionmentioning

confidence: 99%

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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%

Section: Introductionmentioning

confidence: 99%

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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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“…This bound was improved recently for the first time by an algorithm of Gupta et al [10], which runs in n (1.981+o(1))k (randomized) time. The deterministic algorithms side has seen a series of improvements since then [12,24,4]. The fastest algorithm for general edge weights is due to Chekuri et al [4].…”

Section: Introductionmentioning

confidence: 99%

“…The deterministic algorithms side has seen a series of improvements since then [12,24,4]. The fastest algorithm for general edge weights is due to Chekuri et al [4]. It runs in O(mn 2k−3 ) time and is based on a deterministic tree packing result of Thorup [24].…”

Section: Introductionmentioning

confidence: 99%

Faster Minimum k-cut of a Simple Graph

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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We consider the (exact, minimum) k-Cut problem: given a graph and an integer k, delete a minimumweight set of edges so that the remaining graph has at least k connected components. This problem is a natural generalization of the global minimum cut problem, where the goal is to break the graph into k = 2 pieces.Our main result is a (combinatorial) k-Cut algorithm on simple graphs that runs in n (1+o(1))k time for any constant k, improving upon the previously best n (2ω/3+o(1))k time algorithm of Gupta et al. [FOCS'18] and the previously best n (1.981+o(1))k time combinatorial algorithm of Gupta et al. [STOC'19]. For combinatorial algorithms, this algorithm is optimal up to o(1) factors assuming recent hardness conjectures: we show by a straightforward reduction that k-Cut on even a simple graph is as hard as (k − 1)-clique, establishing a lower bound of n (1−o(1))k for k-Cut. This settles, up to lower-order factors, the complexity of k-Cut on a simple graph for combinatorial algorithms. this algorithm is optimal up to o(1) factors assuming recent hardness conjectures: we show by a straightforward reduction that k-Cut on even a simple graph is as hard as (k − 1)-clique, establishing a lower bound of n (1−o(1))k for k-Cut. This settles, up to lower-order factors, the complexity of k-Cut on a simple graph for combinatorial algorithms. We remark that this is the first setting for k-Cut, except the restricted k = 2 case, where the running time has been determined up to o(1) factors.Theorem 1.1 (Main Result). For any parameter k, there is a (combinatorial, randomized) algorithm that computes the k-Cut of a simple graph in k O(k) n (1+o(1))k time.Theorem 1.2 (Lower Bound). Suppose we assume the conjecture that every combinatorial algorithm for k-clique requires n (1−o(1))k time for any constant k. Then, for any constant k, every combinatorial algorithm for k-Cut of a simple graph also requires n (1−o(1))k time. Our TechniquesOur k-Cut algorithm incorporates algorithmic techniques from a wide array of areas, from graph sparsification to fixed-parameter tractability to tree algorithms.Graph Sparsification Our first algorithmic ingredient is the Kawarabayashi-Thorup (KT) sparsification algorithm, which originated from the breakthrough paper of Kawarabayashi and Thorup on the deterministic minimum cut problem [16]. At a high level, given any simple graph G with minimum cut λ, the algorithm contracts G into a multi-graph ofÕ(m/λ) 1 edges so that any minimum cut of G that has at least two vertices on each side gets "preserved" in the contraction. That is, we never contract an edge in any such minimum cut. Kawarabayashi and Thorup used their contraction procedure to provide the firstÕ(m)-time deterministic algorithm for minimum cut of a simple graph. They first applied the contraction to G, obtaining a multi-graph G on m =Õ(m/λ) edges, and then ran theÕ(mλ)-time minimum cut algorithm of Gabow on G, which works for multi-graphs. This covers the case when the minimum cut of G has at least two vertices on each side; the other ca...

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Minimum Violation Vertex Maps and Their Applications to Cut Problems

Kawarabayashi¹,

Xu²

2020

SIAM J. Discrete Math.

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LP Relaxation and Tree Packing for Minimum $k$-cuts

Cited by 5 publications

References 28 publications

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

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

Faster Minimum k-cut of a Simple Graph

Minimum Violation Vertex Maps and Their Applications to Cut Problems

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