Structural Iterative Rounding for Generalized k-Median Problems
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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 + ϵ and 6.387 + ϵ for k-median with outliers and knapsack median, respectively. These improve on the best-known approximation ratio 7.081 + ϵ for both problems <cit.>.
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