Dependent rounding is a useful technique for optimization problems with hard budget constraints. This framework naturally leads to negative correlation properties. However, what if an application naturally calls for dependent rounding on the one hand, and desires positive correlation on the other? More generally, we develop algorithms that guarantee the known properties of dependent rounding, but also have nearly best-possible behavior -near-independence, which generalizes positive correlation -on "small" subsets of the variables. The recent breakthrough of Li & Svensson for the classical kmedian problem has to handle positive correlation in certain dependent-rounding settings, and does so implicitly. We improve upon Li-Svensson's approximation ratio for k-median from 2.732+ to 2.675+ by developing an algorithm that improves upon various aspects of their work. Our dependent-rounding approach helps us improve the dependence of the runtime on the parameter from Li-Svensson's N O(1/ 2 ) to N O((1/ ) log(1/ )) .
Introduction and High-Level DetailsWe consider two notions in combinatorial optimization: a concrete problem (the classical k-median problem) and a formulation of new types of distributions (generalizations of dependent-rounding techniques); the breakthrough of Li & Svensson on the former [24] uses special cases of the latter. We improve the approximation ratio of [24] for the former, and develop efficient samplers for the latter -which, in particular, show that such distributions exist; we then combine the two to improve the run-time of our approximation algorithm for k-median. The ideas developed here also lead to optimal approximations for certain budgeted satisfiability problems, of which the classical budgeted set-cover problem is a special case. We discuss these contributions in further detail in Sections 1.1, 1.2, and 1.3.
In this paper, we give tight approximation algorithms for the k-center and matroid center problems with outliers. Unfairness arises naturally in this setting: certain clients could always be considered as outliers. To address this issue, we introduce a lottery model in which each client j is allowed to submit a parameter p j ∈ [0, 1] and we look for a random solution that covers every client j with probability at least p j . Our techniques include a randomized rounding procedure to round a point inside a matroid intersection polytope to a basis plus at most one extra item such that all marginal probabilities are preserved and such that a certain linear function of the variables does not decrease in the process with probability one.is minimized.In the RKnapCenter problem, each vertex i ∈ V has a weight w i ∈ [0, 1], and the cardinality constraint (i) is replaced by the knapsack constraint: i∈S w i ≤ 1. Similarly, in the RMatCenter problem, the constraint (i) is replaced by a matroid constraint: S must be an independent set of a given matroid M. Here we assume that we have access to the rank oracle of M.
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