An O *(1.84 k ) Parameterized Algorithm for the Multiterminal Cut Problem

Cao, Yixin; Chen, Jianer; Fan, Jia-Hao

doi:10.1007/978-3-642-40164-0_11

Cited by 12 publications

(41 citation statements)

References 23 publications

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“…Lemma 1 [7] If an edge e = (u, v) ∈ G is guaranteed not to be in at least one multiterminal cut C(G) (i.e. P u = P v ), we can contract e and W(G/e) = W(G).…”

Section: Kernelizationmentioning

confidence: 99%

“…Lemma 2 [7] If an edge e = (u, v) ∈ E is guaranteed to be in a minimum multiterminal cut, i.e. there is a minimum multiterminal cut C(G) in which P u = P v , we can delete e from G and C(G − e) is still a valid minimum multiterminal cut.…”

Section: Proofmentioning

confidence: 99%

“…Low-Degree Vertices [7] A non-terminal vertex with no neighbors (IsolatedVertex) can be deleted as there is no incident edge that could affect a cut. For a non-terminal vertex v with only one adjacent edge e = (v, x) (DegreeOne), e can not be part of the minimum multiterminal cut C(G).…”

Section: Local Contractionmentioning

confidence: 99%

See 2 more Smart Citations

Shared-Memory Branch-and-Reduce for Multiterminal Cuts

Henzinger

Noe

Schulz

2020

2020 Proceedings of the Twenty-Second Workshop on Algorithm Engineering and Experiments (ALENEX)

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We introduce the fastest known exact algorithm for the multiterminal cut problem with k terminals. In particular, we engineer existing as well as new data reduction rules. We use the rules within a branch-and-reduce framework and to boost the performance of an ILP formulation. Our algorithms achieve improvements in running time of up to multiple orders of magnitudes over the ILP formulation without data reductions, which has been the de facto standard used by practitioners. This allows us to solve instances to optimality that are significantly larger than was previously possible.

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“…Lemma 1 [7] If an edge e = (u, v) ∈ G is guaranteed not to be in at least one multiterminal cut C(G) (i.e. P u = P v ), we can contract e and W(G/e) = W(G).…”

Section: Kernelizationmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

See 1 more Smart Citation

Shared-Memory Branch-and-Reduce for Multiterminal Cuts

Henzinger

Noe

Schulz

2020

2020 Proceedings of the Twenty-Second Workshop on Algorithm Engineering and Experiments (ALENEX)

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“…1.1. 3 Step III: Reducing k-Cut to Laminar k-cut We now reduce the general k-Cut problem to Laminar k-cut. This reduction is again based on observations about the graph structure in cases where the iterative greedy algorithms do not get a (2 − ε)-approximation.…”

Section: 12mentioning

confidence: 99%

An FPT Algorithm Beating 2-Approximation for k-Cut

Gupta¹,

Lee²,

Li³

2018

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

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In the k-Cut problem, we are given an edge-weighted graph G and an integer k, and have to remove a set of edges with minimum total weight so that G has at least k connected components. Prior work on this problem gives, for all h ∈ [2, k], a (2 − h/k)-approximation algorithm for k-cut that runs in time n O(h) . Hence to get a (2−ε)-approximation algorithm for some absolute constant ε, the best runtime using prior techniques is n O(kε) . Moreover, it was recently shown that getting a (2−ε)-approximation for general k is NP-hard, assuming the Small Set Expansion Hypothesis.If we use the size of the cut as the parameter, an FPT algorithm to find the exact k-Cut is known, but solving the k-Cut problem exactly is W [1]-hard if we parameterize only by the natural parameter of k. An immediate question is: can we approximate k-Cut better in FPT-time, using k as the parameter?We answer this question positively. We show that for some absolute constant ε > 0, there exists a (2 − ε)-approximation algorithm that runs in time 2. This is the first FPT algorithm that is parameterized only by k and strictly improves the 2-approximation.

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“…Together with its close relative Odd Cycle Transversal (OCT), where one deletes vertices instead of edges, Edge Bipartization was one of the first problems shown to admit a fixed-parameter (FPT) algorithm using the technique of iterative compression. In a breakthrough paper [28] that introduces this methodology, Reed et al showed how to solve OCT in time O(3 k · kmn) 1 . In fact, this was the first FPT algorithm for OCT.…”

Section: Introductionmentioning

confidence: 99%

Edge Bipartization Faster than $$2^k$$ 2 k

Pilipczuk

Wrochna

2017

Algorithmica

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In the Edge Bipartization problem one is given an undirected graph G and an integer k, and the question is whether k edges can be deleted from G so that it becomes bipartite. Guo et al. (J Comput Syst Sci 72(8):1386-1396, 2006 proposed an algorithm solving this problem in time O(2 k · m 2 ); today, this algorithm is a textbook example of an application of the iterative compression technique. Despite extensive progress in the understanding of the parameterized complexity of graph separation problems in the recent years, no significant improvement upon this result has been yet reported. We present an algorithm for Edge Bipartization that works in time O(1.977 k · nm), which is the first algorithm with the running time dependence on the parameter better than 2 k . To this end, we combine the general iterative compression strategy of Guo et al. (2006), the technique proposed by Wahlström (in: Proceedings of SODA'14, SIAM, 2014) of using a polynomial-time solvable relaxation in the form of a Valued Constraint Satisfaction Problem to guide a bounded-depth branching algorithm, and an involved Measure&Conquer analysis of the recursion tree.

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An O *(1.84 k ) Parameterized Algorithm for the Multiterminal Cut Problem

Cited by 12 publications

References 23 publications

Shared-Memory Branch-and-Reduce for Multiterminal Cuts

Shared-Memory Branch-and-Reduce for Multiterminal Cuts

An FPT Algorithm Beating 2-Approximation for k-Cut

Edge Bipartization Faster than $$2^k$$ 2 k

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