Structural Iterative Rounding for Generalized $k$-Median Problems

Anupam Gupta; Benjamin Moseley; Rudy Zhou

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arXiv · 2009.00808

Structural Iterative Rounding for Generalized $k$-Median Problems

Anupam Gupta·Benjamin Moseley·Rudy Zhou

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Abstract

This paper considers approximation algorithms for generalized $k$-median problems. This class of problems can be informally described as $k$-median with a constant number of extra constraints, and includes $k$-median with outliers, and knapsack median. Our first contribution is a pseudo-approximation algorithm for generalized $k$-median that outputs a $6.387$-approximate solution, with a constant number of fractional variables. The algorithm builds on the iterative rounding framework introduced by Krishnaswamy, Li, and Sandeep for $k$-median with outliers. The main technical innovation is allowing richer constraint sets in the iterative rounding and taking advantage of the structure of the resulting extreme points. Using our pseudo-approximation algorithm, we give improved approximation algorithms for $k$-median with outliers and knapsack median. This involves combining our pseudo-approximation with pre- and post-processing steps to round a constant number of fractional variables at a small increase in cost. Our algorithms achieve approximation ratios $6.994 + \epsilon$ and $6.387 + \epsilon$ for $k$-median with outliers and knapsack median, respectively. These improve on the best-known approximation ratio $7.081 + \epsilon$ for both problems \cite{DBLP:conf/stoc/KrishnaswamyLS18}.

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BibTeX RIS

Anupam Gupta, Benjamin Moseley, Rudy Zhou. 2020-09-02. Structural Iterative Rounding for Generalized $k$-Median Problems. https://arxiv.org/abs/2009.00808

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