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Rearrangement clustering: Pitfalls, remedies, and applications
 Journal of Machine Learning Research
, 2006
"... Given a matrix of values in which the rows correspond to objects and the columns correspond to features of the objects, rearrangement clustering is the problem of rearranging the rows of the matrix such that the sum of the similarities between adjacent rows is maximized. Referred to by various names ..."
Abstract

Cited by 8 (0 self)
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Given a matrix of values in which the rows correspond to objects and the columns correspond to features of the objects, rearrangement clustering is the problem of rearranging the rows of the matrix such that the sum of the similarities between adjacent rows is maximized. Referred to by various names and reinvented several times, this clustering technique has been extensively used in many fields over the last three decades. In this paper, we point out two critical pitfalls that have been previously overlooked. The first pitfall is deleterious when rearrangement clustering is applied to objects that form natural clusters. The second concerns a similarity metric that is commonly used. We present an algorithm that overcomes these pitfalls. This algorithm is based on a variation of the Traveling Salesman Problem. It offers an extra benefit as it automatically determines cluster boundaries. Using this algorithm, we optimally solve four benchmark problems and a 2,467gene expression data clustering problem. As expected, our new algorithm identifies better clusters than those found by previous approaches in all five cases. Overall, our results demonstrate the benefits of rectifying the pitfalls and exemplify the usefulness of this clustering technique. Our code is available at our websites.
Optimal Toll Design: A Lower Bound Framework for the Asymmetric Traveling Salesman Problem
, 2012
"... We propose a framework of lower bounds for the asymmetric traveling salesman problem (TSP) based on approximating the dynamic programming formulation with different basis vector sets. We discuss how several wellknown TSP lower bounds correspond to intuitive basis vector choices and give an economic ..."
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We propose a framework of lower bounds for the asymmetric traveling salesman problem (TSP) based on approximating the dynamic programming formulation with different basis vector sets. We discuss how several wellknown TSP lower bounds correspond to intuitive basis vector choices and give an economic interpretation wherein the salesman must pay tolls as he travels between cities. We then introduce an exact reformulation that generates a family of successively tighter lower bounds, all solvable in polynomial time. We show that the base member of this family yields a bound greater than or equal to the wellknown HeldKarp bound, obtained by solving the linear programming relaxation of the TSP’s integer programming arcbased formulation.