A simple algorithm for the multiway cut problem

Buchbinder, Niv; Schwartz, Roy; Weizman, Baruch

doi:10.1016/j.orl.2019.09.012

Cited by 4 publications

(4 citation statements)

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“…It is known that Graph-k-Cut is W [1]-hard (and hence not likely to be FPT) parameterized by k [7] while it is FPT when parameterized by k and the solution size [19]. We observed, via a simple reduction from a result of Marx on vertex separators [24], that Hypergraph-k-Cut is W [1] hard even when parameterized by k and the solution size. This also demonstrates that Hypergraph-k-Cut differs in complexity from Graph-k-Cut.…”

Section: Other Related Workmentioning

confidence: 89%

“…These problems are NP-Hard even for k = 3 and the main focus has been on approximation algorithms. We refer the reader to [1,3,8,34] for further references. We mention that for non-negative f and fixed k, the best approximation algorithms for Submod-k-Part and Sym-Submod-k-Part are via the terminal versions; a (1.5 − 1/k) for Sym-Submod-k-Part and a 2(1 − 1/k)-approximation for Submod-k-Part [3,8].…”

Section: Other Related Workmentioning

confidence: 99%

“…It is not hard to see that Graph-k-Cut is a special case of Sym-Submod-k-Part. However, Hypergraph-k-Cut is not a special case of Sym-Submod-k-Part even though the hypergraph cut function is itself symmetric; 1 as observed in [26], one can reduce Hypergraph-k-Cut to Submod-k-Part. Submod-k-Part and Sym-Submod-k-Part are very general problems.…”

Section: Introductionmentioning

confidence: 99%

“…Thorup [32] showed that tree packings can also be used to obtain a polynomial-time algorithm for Graph-k-Cut. His algorithm runs in deterministic n 2k+O (1) time; his approach was clarified in [5] via an LP relaxation and this also resulted in a slight improvement in the run-time and currently yields the fastest deterministic algorithm. We defer discussion of approximation algorithms for Graph-k-Cut when k is part of the input to the related work section.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Hypergraph k-Cut in Randomized Polynomial Time

Chandrasekaran¹,

Xu²,

Yu³

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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In the hypergraph k-cut problem, the input is a hypergraph, and the goal is to find a smallest subset of hyperedges whose removal ensures that the remaining hypergraph has at least k connected components. This problem is known to be at least as hard as the densest k-subgraph problem when k is part of the input (Chekuri-Li, 2015). We present a randomized polynomial time algorithm to solve the hypergraph k-cut problem for constant k. Our algorithm solves the more general hedge k-cut problem when the subgraph induced by every hedge has a constant number of connected components. In the hedge k-cut problem, the input is a hedgegraph specified by a vertex set and a disjoint set of hedges, where each hedge is a subset of edges defined over the vertices. The goal is to find a smallest subset of hedges whose removal ensures that the number of connected components in the remaining underlying (multi-)graph is at least k. Our algorithm is based on random contractions akin to Karger's min cut algorithm. Our main technical contribution is a distribution over the hedges (hyperedges) so that random contraction of hedges (hyperedges) chosen from the distribution succeeds in returning an optimum solution with large probability.

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Section: Other Related Workmentioning

confidence: 89%