14 research outputs found
Evolutionary trees: an integer multicommodity max-flow-min-cut theorem
In biomathematics, the extensions of a leaf-colouration of a binary tree to the whole vertex set with minimum number of colour-changing edges are extensively studied. Our paper generalizes the problem for trees; algorithms and a Menger-type theorem are presented. The LP dual of the problem is a multicommodity flow problem, for which a max-flow-min-cut theorem holds. The problem that we solve is an instance of the NP-hard multiway cut problem
Evaluation of ILP-based approaches for partitioning into colorful components
The NP-hard Colorful Components problem is a graph partitioning problem on vertex-colored graphs. We identify a new application of Colorful Components in the correction of Wikipedia interlanguage links, and describe and compare three exact and two heuristic approaches. In particular, we devise two ILP formulations, one based on Hitting Set and one based on Clique Partition. Furthermore, we use the recently proposed implicit hitting set framework [Karp, JCSS 2011; Chandrasekaran et al., SODA 2011] to solve Colorful Components. Finally, we study a move-based and a merge-based heuristic for Colorful Components. We can optimally solve Colorful Components for Wikipedia link correction data; while the Clique Partition-based ILP outperforms the other two exact approaches, the implicit hitting set is a simple and competitive alternative. The merge-based heuristic is very accurate and outperforms the move-based one. The above results for Wikipedia data are confirmed by experiments with synthetic instances
Nonlinear Formations and Improved Randomized Approximation Algorithms for Multiway and Multicut Problems
We introduce nonlinear formulations of the multiway cut and multicut problems. By simple linearizations of these formulations we derive several well known formulations and valid inequalities as well as several new ones. Through these formulations we establish a connection between the multiway cut and the maximum weighted independent set problem that leads to the study of the tightness of several LP formulations for the multiway cut problem through the theory of perfect graphs. We also introduce a new randomized rounding argument to study the worst case bound of these formulations, obtaining a new bound of 2a(H)(1 - ) for the multicut problem, where ac(H) is the size of a maximum independent set in the demand graph H
Nonlinear formulations and improved randomized approximation algorithms for multiway and multicut problems
Cover title.Includes bibliographical references (p. 21-22).D. Bertsimas, C. Teo and R. Vohra
A Generalized Framework for Agglomerative Clustering of Signed Graphs applied to Instance Segmentation
We propose a novel theoretical framework that generalizes algorithms for
hierarchical agglomerative clustering to weighted graphs with both attractive
and repulsive interactions between the nodes. This framework defines GASP, a
Generalized Algorithm for Signed graph Partitioning, and allows us to explore
many combinations of different linkage criteria and cannot-link constraints. We
prove the equivalence of existing clustering methods to some of those
combinations, and introduce new algorithms for combinations which have not been
studied. An extensive comparison is performed to evaluate properties of the
clustering algorithms in the context of instance segmentation in images,
including robustness to noise and efficiency. We show how one of the new
algorithms proposed in our framework outperforms all previously known
agglomerative methods for signed graphs, both on the competitive CREMI 2016 EM
segmentation benchmark and on the CityScapes dataset.Comment: 19 pages, 8 figures, 6 table
Algorithms for Cut Problems on Trees
We study the {\sc multicut on trees} and the {\sc generalized multiway Cut on
trees} problems. For the {\sc multicut on trees} problem, we present a
parameterized algorithm that runs in time , where is the positive root of the polynomial
. This improves the current-best algorithm of Chen et al. that runs
in time . For the {\sc generalized multiway cut on trees}
problem, we show that this problem is solvable in polynomial time if the number
of terminal sets is fixed; this answers an open question posed in a recent
paper by Liu and Zhang. By reducing the {\sc generalized multiway cut on trees}
problem to the {\sc multicut on trees} problem, our results give a
parameterized algorithm that solves the {\sc generalized multiway cut on trees}
problem in time , where time