A new approach to the maximum-flow problem
Journal of the ACM (JACM)
A fast parametric maximum flow algorithm and applications
SIAM Journal on Computing
Approximate counting, uniform generation and rapidly mixing Markov chains
Information and Computation
The Markov chain Monte Carlo method: an approach to approximate counting and integration
Approximation algorithms for NP-hard problems
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
Normalized Cuts and Image Segmentation
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Min-max Cut Algorithm for Graph Partitioning and Data Clustering
ICDM '01 Proceedings of the 2001 IEEE International Conference on Data Mining
Multiclass Spectral Clustering
ICCV '03 Proceedings of the Ninth IEEE International Conference on Computer Vision - Volume 2
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Graph Partitioning by Spectral Rounding: Applications in Image Segmentation and Clustering
CVPR '06 Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition - Volume 1
"Ratio Regions": A Technique for Image Segmentation
ICPR '96 Proceedings of the 13th International Conference on Pattern Recognition - Volume 2
A survey of kernel and spectral methods for clustering
Pattern Recognition
The Pseudoflow Algorithm: A New Algorithm for the Maximum-Flow Problem
Operations Research
Polynomial Time Algorithms for Ratio Regions and a Variant of Normalized Cut
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Nearly-m log n Time Solver for SDD Linear Systems
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
IEEE Transactions on Information Theory - Part 1
Linear-time encodable and decodable error-correcting codes
IEEE Transactions on Information Theory - Part 1
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A general form of minimizing the Rayleigh ratio on discrete variables is shown here, for the first time, to be polynomial time solvable. This is significant because major problems in clustering, partitioning, and imaging can be presented as the Rayleigh ratio minimization on discrete variables and an orthogonality constraint. These challenging problems are modeled as the normalized cut problem, the graph expander ratio problem, the Cheeger constant problem, or the conductance problem, all of which are NP-hard. These problems have traditionally been solved, heuristically, using the “spectral technique.” A unified framework is provided here whereby all these problems are formulated as a constrained minimization form of a quadratic ratio, referred to here as the Rayleigh ratio. The quadratic ratio is to be minimized on discrete variables and a single sum constraint that we call the balance or orthogonality constraint. When the discreteness constraints on the variables are disregarded, the resulting continuous relaxation is solved by the spectral method. It is shown here that the Rayleigh ratio minimization subject to the discreteness constraints requiring each variable to assume one of two values in {-b,1} is solvable in strongly polynomial time, equivalent to a single minimum s,t cut algorithm on a graph of same size as the input graph, for any nonnegative value of b. This discrete form for the Rayleigh ratio problem was often assumed to be NP-hard. Not only is it shown here that the discrete Rayleigh ratio problem is polynomial time solvable, but also the algorithm is more efficient than the spectral algorithm. Furthermore, an experimental study demonstrates that the new algorithm provides in practice an improvement, often dramatic, on the quality of the results of the spectral method, both in terms of approximating the true optimum of the Rayleigh ratio problem on both the discrete variables and the balance constraint, and in terms of the subjective partition quality. A further contribution here is the introduction of a problem, the quantity-normalized cut, generalizing all the Rayleigh ratio problems. The discrete version of that problem is also solved with the efficient algorithm presented. This problem is shown, in a companion paper, to enable the modeling of features essential to clustering that are valuable in practical applications.