Optimal 3-terminal cuts and linear programming

Cheung, Kevin K. H.; Cunningham, William H.; Tang, Lawrence

doi:10.1007/s10107-005-0668-2

Cited by 23 publications

(7 citation statements)

References 9 publications

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“…In particular, (Cȃlinescu et al, 2000) give a linear programming relaxation resulting in an efficient algorithm guaranteed to find a cut size no larger than ( 3 2 − 1 k ) times the optimal size. This result has since been slightly improved for the general case (Karger et al, 2004), and has been reduced to within a factor of 12 11 of the optimal cut size for the special case of k = 3 (Cheung et al, 2005;Karger et al, 2004). Note that these are worst-case guarantees, and these algorithms often produce optimal solutions in practice.…”

Section: A Learning Bias Toward Small Cutsmentioning

confidence: 76%

An analysis of graph cut size for transductive learning

Hanneke

2006

Proceedings of the 23rd International Conference on Machine Learning - ICML '06

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I consider the setting of transductive learning of vertex labels in graphs, in which a graph with n vertices is sampled according to some unknown distribution; there is a true labeling of the vertices such that each vertex is assigned to exactly one of k classes, but the labels of only some (random) subset of the vertices are revealed to the learner. The task is then to find a labeling of the remaining (unlabeled) vertices that agrees as much as possible with the true labeling. Several existing algorithms are based on the assumption that adjacent vertices are usually labeled the same. In order to better understand algorithms based on this assumption, I derive data-dependent bounds on the fraction of mislabeled vertices, based on the number (or total weight) of edges between vertices differing in predicted label (i.e., the size of the cut).

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Section: A Learning Bias Toward Small Cutsmentioning

confidence: 76%

An analysis of graph cut size for transductive learning

Hanneke

2006

Proceedings of the 23rd International Conference on Machine Learning - ICML '06

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“…Cȃlinescu et al [5] developed an elegant (1.5 − 1 r )-approximation algorithm that solves an LP to embed a graph into an r-simplex, then cuts the simplex using side-parallel cuts (hyperplanes parallel to the faces of the simplex) to induce a multiway cut in the graph. This same approach was improved by Karger et al [19] to obtain a guarantee of 1.3438 (which can be slightly improved for small values of k), and Karger et al as well as Cheung et al [7] independently obtained a guarantee of 12/11 for the special case of r = 3.…”

Section: Literature Reviewmentioning

confidence: 82%

Approximation Algorithms for k-hurdle Problems

et al. 2010

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The polynomial-time solvable k-hurdle problem is a natural generalization of the classical s-t minimum cut problem where we must select a minimum-cost subset S of the edges of a graph such that |p ∩ S| ≥ k for every s-t path p. In this paper, we describe a set of approximation algorithms for "k-hurdle" variants of the NP-hard multiway cut and multicut problems. For the k-hurdle multiway cut problem with r terminals, we give two results, the first being a pseudo-approximation algorithm that outputs a (k − 1)-hurdle solution whose cost is at most that of an optimal solution for k hurdles. Secondly, we provide a 2(1 − 1 r )-approximation algorithm based on rounding the solution of a linear program, for which we give a simple randomized half-integrality proof that works for both edge and vertex k-hurdle multiway cuts that generalizes the half-integrality results of Garg et al. for the vertex multiway cut problem. We also describe an approximation-preserving reduction from vertex cover as evidence that it may be difficult to achieve a better approximation ratio than 2(1 − 1 r ). For the k-hurdle multicut problem in an n-vertex graph, we provide an algorithm that, for any constant ε > 0, outputs a (1 − ε)k -hurdle solution of cost at most O(log n) times that of an optimal k-hurdle solution, and we obtain a 2-approximation algorithm for trees.

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“…It is well-known that k-Way-Cut is NP-hard [7]. For k = 3, a 12/11-approximation is known [5,11], while for constant k, the current-best approximation factor is 1.2975 due to Sharma and Vondrák [19]. These results are based on an LP-relaxation proposed by Cȃlinescu, Karloff and Rabani [6], known as the CKR relaxation.…”

Section: Related Workmentioning

confidence: 99%

Beating the 2-approximation factor for global bicut

et al. 2018

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In the fixed-terminal bicut problem, the input is a directed graph with two specified nodes and the goal is to find a smallest subset of edges whose removal ensures that the two specified nodes cannot reach each other. In the global bicut problem, the input is a directed graph and the goal is to find a smallest subset of edges whose removal ensures that there exist two nodes that cannot reach each other. Fixed-terminal bicut and global bicut are natural extensions of {s, t}-min cut and global min-cut respectively, from undirected graphs to directed graphs. Fixed-terminal bicut is NP-hard, admits a simple 2-approximation, and does not admit a (2 −)-approximation for any constant > 0 assuming the unique games conjecture. In this work, we show that global bicut admits a (2 − 1/448)approximation, thus improving on the approximability of the global variant in comparison to the fixed-terminal variant.

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Optimal 3-terminal cuts and linear programming

Cited by 23 publications

References 9 publications

An analysis of graph cut size for transductive learning

An analysis of graph cut size for transductive learning

Approximation Algorithms for k-hurdle Problems

Beating the 2-approximation factor for global bicut

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