
Exact and Approximation Algorithms for Sparse PCA
Sparse PCA (SPCA) is a fundamental model in machine learning and data an...
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Outliers Detection Is Not So Hard: Approximation Algorithms for Robust Clustering Problems Using Local Search Techniques
In this paper, we consider two types of robust models of the kmedian/k...
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Iterated Tabu Search Algorithm for Packing Unequal Circles in a Circle
This paper presents an Iterated Tabu Search algorithm (denoted by ITSPU...
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Instance Scale, Numerical Properties and Design of Metaheuristics: A Study for the Facility Location Problem
Metaheuristics are known to be strong in solving largescale instances o...
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Local Search Yields a PTAS for kMeans in Doubling Metrics
The most well known and ubiquitous clustering problem encountered in nea...
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A Tight Extremal Bound on the Lovász Cactus Number in Planar Graphs
A cactus graph is a graph in which any two cycles are edgedisjoint. We ...
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Computing Maximum Entropy Distributions Everywhere
We study the problem of computing the maximum entropy distribution with ...
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Best Principal Submatrix Selection for the Maximum Entropy Sampling Problem: Scalable Algorithms and Performance Guarantees
This paper studies a classic maximum entropy sampling problem (MESP), which aims to select the most informative principal submatrix of a prespecified size from a covariance matrix. MESP has been widely applied to many areas, including healthcare, power system, manufacturing and data science. By investigating its Lagrangian dual and primal characterization, we derive a novel convex integer program for MESP and show that its continuous relaxation yields a nearoptimal solution. The results motivate us to study an efficient sampling algorithm and develop its approximation bound for MESP, which improves the bestknown bound in literature. We then provide an efficient deterministic implementation of the sampling algorithm with the same approximation bound. By developing new mathematical tools for the singular matrices and analyzing the Lagrangian dual of the proposed convex integer program, we investigate the widelyused local search algorithm and prove its firstknown approximation bound for MESP. The proof techniques further inspire us with an efficient implementation of the local search algorithm. Our numerical experiments demonstrate that these approximation algorithms can efficiently solve mediumsized and largescale instances to nearoptimality. Our proposed algorithms are coded and released as opensource software. Finally, we extend the analyses to the AOptimal MESP (AMESP), where the objective is to minimize the trace of the inverse of the selected principal submatrix.
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