Isolation branching: a branch and bound algorithm for the k-terminal cut problem

Velednitsky, Mark

doi:10.1007/s10878-020-00534-y

Cited by 2 publications

(2 citation statements)

References 15 publications

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“…In summary, while our proposed algorithm only works on more stable instances, under reasonable assumptions it is at least k times faster than solving the CKR relaxation. In practical instances, the speed-up may be several orders of magnitude (Velednitsky and Hochbaum 2018). This analysis does not account for the possibility of solving the CKR relaxation by a means that avoids solving a systems of equations or inverting a matrix.…”

Section: Practical Usabilitymentioning

confidence: 99%

Solving $$(k-1)$$-stable instances of k-terminal cut with isolating cuts

Velednitsky

2021

J Comb Optim

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The problem, also known as the multiterminal cut problem, is defined on an edge-weighted graph with k distinct vertices called “terminals.” The goal is to remove a minimum weight collection of edges from the graph such that there is no path between any pair of terminals. The problem is APX-hard. Isolating cuts are minimum cuts which separate one terminal from the rest. The union of all the isolating cuts, except the largest, is a $$(2-2/k)$$ ( 2 - 2 / k ) -approximation to the optimal k-terminal cut. An instance of is $$\gamma $$ γ -stable if edges in the cut can be multiplied by up to $$\gamma $$ γ without changing the unique optimal solution. In this paper, we show that, in any $$(k-1)$$ ( k - 1 ) -stable instance of , the source sets of the isolating cuts are the source sets of the unique optimal solution to that instance. We conclude that the $$(2-2/k)$$ ( 2 - 2 / k ) -approximation algorithm returns the optimal solution on $$(k-1)$$ ( k - 1 ) -stable instances. Ours is the first result showing that this $$(2-2/k)$$ ( 2 - 2 / k ) -approximation is an exact optimization algorithm on a special class of graphs. We also show that our $$(k-1)$$ ( k - 1 ) -stability result is tight. We construct $$(k-1-\epsilon )$$ ( k - 1 - ϵ ) -stable instances of the problem which only have trivial isolating cuts: that is, the source set of the isolating cuts for each terminal is just the terminal itself. Thus, the $$(2-2/k)$$ ( 2 - 2 / k ) -approximation does not return an optimal solution.

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Section: Practical Usabilitymentioning

confidence: 99%

Solving $$(k-1)$$-stable instances of k-terminal cut with isolating cuts

Velednitsky

2021

J Comb Optim

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“…Recent work has shown that the multi-way partitioning problem can be solved in polynomial time on special graphs [30] and that a tailored branch-and-bound algorithm can solve instances with tens of thousands of nodes in just seconds [29].…”

Section: Minimum Cut Formulation 71 Backgroundmentioning

confidence: 99%

Anomaly Detection and Correction in Large Labeled Bipartite Graphs

Darling¹,

Velednitsky²

2018

Preprint

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Binary classification problems can be naturally modeled as bipartite graphs, where we attempt to classify right nodes based on their left adjacencies. We consider the case of labeled bipartite graphs in which some labels and edges are not trustworthy. Our goal is to reduce noise by identifying and fixing these labels and edges.We first propose a geometric technique for generating random graph instances with untrustworthy labels and analyze the resulting graph properties. We focus on generating graphs which reflect real-world data, where degree and label frequencies follow power law distributions.We review several algorithms for the problem of detection and correction, proposing novel extensions and making observations specific to the bipartite case. These algorithms range from math programming algorithms to discrete combinatorial algorithms to Bayesian approximation algorithms to machine learning algorithms.We compare the performance of all these algorithms using several metrics and, based on our observations, identify the relative strengths and weaknesses of each individual algorithm.

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Isolation branching: a branch and bound algorithm for the k-terminal cut problem

Cited by 2 publications

References 15 publications

Solving $$(k-1)$$-stable instances of k-terminal cut with isolating cuts

Solving $$(k-1)$$-stable instances of k-terminal cut with isolating cuts

Anomaly Detection and Correction in Large Labeled Bipartite Graphs

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