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Combinatorial Persistency Criteria for Multicut and Max-Cut

Jan-Hendrik Lange, Bjoern Andres, Paul Swoboda
2019 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)  
We propose persistency criteria for the multicut and max-cut problem as well as fast combinatorial routines to verify them.  ...  The more advanced ones rely on fast algorithms for upper and lower bounds for the respective cut problems and max-flow techniques for auxiliary min-cut problems.  ...  The MUL-TICUT problem (a.k.a. correlation clustering) and MAX-CUT problem are arguably among the most well-known combinatorial optimization problems for partitioning graphs.  ... 
doi:10.1109/cvpr.2019.00625 dblp:conf/cvpr/LangeAS19 fatcat:hysiqvg6rvasvf6q7vng7ezp5y

Parallel Multicut Segmentation via Dual Decomposition [chapter]

Julian Yarkony, Thorsten Beier, Pierre Baldi, Fred A. Hamprecht
2015 Lecture Notes in Computer Science  
We propose a new outer relaxation of the multicut polytope, along with a dual decomposition approach for correlation clustering and multicut segmentation, for general graphs.  ...  An optimal reparameterization is found using subgradients and affords a new characterization of the trivial LP relaxation of the multicut problem, as well as informed decoding heuristics.  ...  The multicut problem is fundamental to machine learning in the guises of correlation clustering (8; 5) and image segmentation (22; 1; 2; 3; 12) .  ... 
doi:10.1007/978-3-319-17876-9_4 fatcat:4j2us6v6cvcmvbcbik33flx4i4

Local Guarantees in Graph Cuts and Clustering [chapter]

Moses Charikar, Neha Gupta, Roy Schwartz
2017 Lecture Notes in Computer Science  
Correlation Clustering is an elegant model that captures fundamental graph cut problems such as Min s − t Cut, Multiway Cut, and Multicut, extensively studied in combinatorial optimization.  ...  This naturally gives rise to a family of basic min-max graph cut problems.  ...  Min Max Disagreements also gives rise to Min Max Multiway Cut and Min Max Multicut, defined similarly; no approximation is known for these.  ... 
doi:10.1007/978-3-319-59250-3_12 fatcat:ojmqzw75ujf3lk4xl4jc2o5rym

Planar Ultrametric Rounding for Image Segmentation [article]

Julian Yarkony, Charless C. Fowlkes
2015 arXiv   pre-print
We study the problem of hierarchical clustering on planar graphs. We formulate this in terms of an LP relaxation of ultrametric rounding.  ...  Computing minimum-weight multicuts (also known as correlation clustering) is NP hard even in the case of planar graphs [6] .  ...  ( Zγ m − Zγ l ) = θ +l · (Zγ l + max m≥l (Zγ m − Zγ l )) + θ −l · (Zγ l − max(0, Zγ l − 1)) = θ +l · max m≥l Zγ m + θ −l · min(1, Zγ l ) ≥ θ +l · min(1, max m≥l Zγ m ) + θ −l · min(1, Zγ l ) ≥ θ +l  ... 
arXiv:1507.02407v3 fatcat:xjx22u5zp5cglmbaxsagra7mfy

Combinatorial persistency criteria for multicut and max-cut [article]

Jan-Hendrik Lange, Bjoern Andres, Paul Swoboda
2018 arXiv   pre-print
We propose persistency criteria for the multicut and max-cut problem as well as fast combinatorial routines to verify them.  ...  The more advanced ones rely on fast algorithms for upper and lower bounds for the respective cut problems and max-flow techniques for auxiliary min-cut problems.  ...  The MULTICUT problem (a.k.a. correlation clustering) and MAX-CUT problem are arguably among the most well-known combinatorial optimization problems for partitioning graphs.  ... 
arXiv:1812.01426v1 fatcat:eal5xmlvkzbhzn2r744w7hemra

Correlation clustering in general weighted graphs

Erik D. Demaine, Dotan Emanuel, Amos Fiat, Nicole Immorlica
2006 Theoretical Computer Science  
Correlation clustering was introduced by Bansal et al. motivated by both document clustering and agnostic learning.  ...  In contrast to most clustering problems, correlation clustering specifies neither the desired number of clusters nor a distance threshold for clustering; both of these parameters are effectively chosen  ...  Their paper also inspires our O(r 3 ) approximation which uses a rounding technique introduced by Tardos and Vazirani [27] in a paper on max-flow min-multicut ratio and based on a lemma of Klein et al  ... 
doi:10.1016/j.tcs.2006.05.008 fatcat:xscq7fvjsjejxkrk5ybvhwkshe

Flow-Partitionable Signed Graphs [article]

Jan-Hendrik Lange
2020 arXiv   pre-print
In other words, flow-partitionable signed graphs satisfy an exact max-multiflow-min-multicut relation in the associated instances of minimum multicut.  ...  The NP-hard problem of correlation clustering is to partition a signed graph such that the number of conflicts between the partition and the signature of the graph is minimized.  ...  By reduction to minimum multicut, the class of flow-partitionable signed graphs define instances for which an exact max-multiflow-min-multicut relation holds.  ... 
arXiv:2005.01536v1 fatcat:2st2tly7cndljfusryl4tzl7eu

Structured Prediction Problem Archive [article]

Paul Swoboda, Andrea Hornakova, Paul Roetzer, Bogdan Savchynskyy, Ahmed Abbas
2022 arXiv   pre-print
Multicut The multicut problem [CR93] (also known as correlation clustering [DEFI06] ) is to cluster graph nodes based on edge preferences. Definition 3 (Multicut).  ...  CGC [BKK + 14]: Cut, Glue & Cut, a heuristic that alternatingly bipartitions the graph and exchanges nodes in pairs of clusters via max-cut computed by a reduction to perfect matching.  ... 
arXiv:2202.03574v2 fatcat:n5ez3attlveyvamyz5l4sdgozi

RAMA: A Rapid Multicut Algorithm on GPU [article]

Ahmed Abbas, Paul Swoboda
2022 arXiv   pre-print
We propose a highly parallel primal-dual algorithm for the multicut (a.k.a. correlation clustering) problem, a classical graph clustering problem widely used in machine learning and computer vision.  ...  Our algorithm consists of three steps executed recursively: (1) Finding conflicted cycles that correspond to violated inequalities of the underlying multicut relaxation, (2) Performing message passing  ...  The multicut problem [15] (also known as correlation clustering [10] ) is a popular approach to decompose a graph into an arbitrary number of clusters based on affinites between nodes.  ... 
arXiv:2109.01838v3 fatcat:co2zowc42fcwtfrp666gxcvezq

A Message Passing Algorithm for the Minimum Cost Multicut Problem

Paul Swoboda, Bjoern Andres
2017 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)  
We propose a dual decomposition and linear program relaxation of the NP-hard minimum cost multicut problem.  ...  Unlike other polyhedral relaxations of the multicut polytope, it is amenable to efficient optimization by message passing.  ...  The authors would like to thank Vladimir Kolmogorov for helpful discussions and gratefully acknowledge Fred Hamprecht's and Thorsten Beier's help in comparing OpenGM's multicut solvers.  ... 
doi:10.1109/cvpr.2017.530 dblp:conf/cvpr/SwobodaA17 fatcat:achpvdfh7beypd56ep2z2k77hm

Decomposition of Trees and Paths via Correlation [article]

Jan-Hendrik Lange, Bjoern Andres
2017 arXiv   pre-print
We characterize several classes of facets of the combinatorial polytope associated with a formulation of this clustering problem in terms of lifted multicuts.  ...  We study the problem of decomposing (clustering) a tree with respect to costs attributed to pairs of nodes, so as to minimize the sum of costs for those pairs of nodes that are in the same component (cluster  ...  As any decomposition of a graph is characterized by the mathematical notion of a multicut, the study of multicuts is closely related to correlation clustering.  ... 
arXiv:1706.06822v2 fatcat:nsm2yilpkvefzkzxyn7ctq665m

Partial Optimality and Fast Lower Bounds for Weighted Correlation Clustering

Jan-Hendrik Lange, Andreas Karrenbauer, Bjoern Andres
2018 International Conference on Machine Learning  
Weighted correlation clustering is hard to solve and hard to approximate for general graphs. Its applications in network analysis and computer vision call for efficient algorithms.  ...  G, w 0 and w 1 is the optimization problem min Π e∈E 0 Π w 0 e + e∈E 1 Π w 1 e . (3) Minimum Cost Multicut Weighted correlation clustering is commonly stated in the form of a binary program whose feasible  ...  Problem Formulations Weighted Correlation Clustering Weighted correlation clustering is a combinatorial optimization problem whose feasible solutions are all clusterings of a graph.  ... 
dblp:conf/icml/LangeKA18 fatcat:zfdjdq27rvbjheongmresl5rxa

Estimating the Robustness of Classification Models by the Structure of the Learned Feature-Space [article]

Kalun Ho, Franz-Josef Pfreundt, Janis Keuper, Margret Keuper
2021 arXiv   pre-print
We introduce robustness indicators which are obtained via unsupervised clustering of latent representations from a trained classifier and show very high correlations to the model performance on corrupted  ...  Thus, the purity score is higher than the cluster accuracy (80% vs. 73%). min y∈{0,1} E e∈E w e y e (2) s.t.  ...  Multicut Clustering. The Minimum Cost Multicut Problem is a graph based clustering approach.  ... 
arXiv:2106.12303v2 fatcat:l7kpjljeezexhdx7tdote4klzq

Robust Correlation Clustering

Devvrit, Ravishankar Krishnaswamy, Nived Rajaraman, Michael Wagner
2019 International Workshop on Approximation Algorithms for Combinatorial Optimization  
Correlation clustering is useful when we have (only) qualitative information about the similarity or dissimilarity of pairs of points, and Robust-Correlation-Clustering equips this model with the capability  ...  This generalizes the classical Correlation-Clustering problem which is the special case when m = 0.  ...  problem on general graphs is equivalent to Minimum-Multicut on general graphs in [14] ), and running the best known approximation to Minimum-Multicut to get O(log n) approximations to Correlation-Clustering  ... 
doi:10.4230/lipics.approx-random.2019.33 dblp:conf/approx/DevvritKR19 fatcat:vgzp4fepezamhbsg3ety6ap6dm

Next Generation Multicuts for Semi-Planar Graphs [article]

Julian Yarkony
2015 arXiv   pre-print
We study the problem of multicut segmentation. We introduce modified versions of the Semi-PlanarCC based on bounding Lagrange multipliers. We apply our work to natural image segmentation.  ...  Eq 10 ≥ min X∈M CU T θX +λ(X − 1) +ψ(1 −X) = min X∈M CU T −λ1 +ψ1 + (θ +λ − ψS)X = −λ1 +ψ1 + min X∈M CU T (θ +λ − ψS)X (11) Solving for minX ∈M CU T (θ +λ − ψS) is exactly Planar Correlation Clustering  ...  We then find violated paths in the primal solution X = min(1,Ẑγ + κ) via shortest path calculation [1] between nodes in each pair f ∈ F .  ... 
arXiv:1511.01994v1 fatcat:tdgnh3oy3vgrpb47dpwyn33zpe
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