Improved Approximation Algorithms for Matroid and Knapsack Median Problems and Applications

We consider the {\em matroid median} problem \cite{KrishnaswamyKNSS11}, wherein we are given a set of facilities with opening costs and a matroid on the facility-set, and clients with demands and connection costs, and we seek to open an independent set of facilities and assign clients to open facilities so as to minimize the sum of the facility-opening and client-connection costs. We give a simple 8-approximation algorithm for this problem based on LP-rounding, which improves upon the 16-approximation in \cite{KrishnaswamyKNSS11}. Our techniques illustrate that much of the work involved in the rounding algorithm of in \cite{KrishnaswamyKNSS11} can be avoided by first converting the LP solution to a half-integral solution, which can then be rounded to an integer solution using a simple uncapacitated-facility-location (UFL) style clustering step. We illustrate the power of these ideas by deriving: (a) a 24-approximation algorithm for matroid median with penalties, which is a vast improvement over the 360-approximation obtained in \cite{KrishnaswamyKNSS11}; and (b) an 8-approximation for the {\em two-matroid median} problem, a generalization of matroid median that we introduce involving two matroids. We show that a variety of seemingly disparate facility-location problems considered in the literaturedata placement problem, mobile facility location, $k$-median forest, metric uniform minimum-latency UFLin fact reduce to the matroid median or two-matroid median problems, and thus obtain improved approximation guarantees for all these problems. Our techniques also yield an improvement for the knapsack median problem.