Matroid and Knapsack Center Problems

In the classic $k$-center problem, we are given a metric graph, and the objective is to open $k$ nodes as centers such that the maximum distance from any vertex to its closest center is minimized. In this paper, we consider two important generalizations of $k$-center, the matroid center problem and the knapsack center problem. Both problems are motivated by recent content distribution network applications. Our contributions can be summarized as follows: 1. We consider the matroid center problem in which the centers are required to form an independent set of a given matroid. We show this problem is NP-hard even on a line. We present a 3-approximation algorithm for the problem on general metrics. We also consider the outlier version of the problem where a given number of vertices can be excluded as the outliers from the solution. We present a 7-approximation for the outlier version. 2. We consider the (multi-)knapsack center problem in which the centers are required to satisfy one (or more) knapsack constraint(s). It is known that the knapsack center problem with a single knapsack constraint admits a 3-approximation. However, when there are at least two knapsack constraints, we show this problem is not approximable at all. To complement the hardness result, we present a polynomial time algorithm that gives a 3-approximate solution such that one knapsack constraint is satisfied and the others may be violated by at most a factor of $1+\epsilon$. We also obtain a 3-approximation for the outlier version that may violate the knapsack constraint by $1+\epsilon$.Comment: A preliminary version of this paper is accepted to IPCO 201

Approximation algorithms for stochastic clustering

We consider stochastic settings for clustering, and develop provably-good approximation algorithms for a number of these notions. These algorithms yield better approximation ratios compared to the usual deterministic clustering setting. Additionally, they offer a number of advantages including clustering which is fairer and has better long-term behavior for each user. In particular, they ensure that *every user* is guaranteed to get good service (on average). We also complement some of these with impossibility results

Constant Approximation for kk-Median and kk-Means with Outliers via Iterative Rounding

In this paper, we present a new iterative rounding framework for many clustering problems. Using this, we obtain an $(\alpha_1 + \epsilon \leq 7.081 + \epsilon)$-approximation algorithm for $k$-median with outliers, greatly improving upon the large implicit constant approximation ratio of Chen [Chen, SODA 2018]. For $k$-means with outliers, we give an $(\alpha_2+\epsilon \leq 53.002 + \epsilon)$-approximation, which is the first $O(1)$-approximation for this problem. The iterative algorithm framework is very versatile; we show how it can be used to give $\alpha_1$- and $(\alpha_1 + \epsilon)$-approximation algorithms for matroid and knapsack median problems respectively, improving upon the previous best approximations ratios of $8$ [Swamy, ACM Trans. Algorithms] and $17.46$ [Byrka et al, ESA 2015]. The natural LP relaxation for the $k$-median/$k$-means with outliers problem has an unbounded integrality gap. In spite of this negative result, our iterative rounding framework shows that we can round an LP solution to an almost-integral solution of small cost, in which we have at most two fractionally open facilities. Thus, the LP integrality gap arises due to the gap between almost-integral and fully-integral solutions. Then, using a pre-processing procedure, we show how to convert an almost-integral solution to a fully-integral solution losing only a constant-factor in the approximation ratio. By further using a sparsification technique, the additive factor loss incurred by the conversion can be reduced to any $\epsilon > 0$

Interpolating between k-Median and k-Center: Approximation Algorithms for Ordered k-Median

We consider a generalization of k-median and k-center, called the ordered k-median problem. In this problem, we are given a metric space (D,{c_{ij}}) with n=|D| points, and a non-increasing weight vector w in R_+^n, and the goal is to open k centers and assign each point j in D to a center so as to minimize w_1 *(largest assignment cost)+w_2 *(second-largest assignment cost)+...+w_n *(n-th largest assignment cost). We give an (18+epsilon)-approximation algorithm for this problem. Our algorithms utilize Lagrangian relaxation and the primal-dual schema, combined with an enumeration procedure of Aouad and Segev. For the special case of {0,1}-weights, which models the problem of minimizing the l largest assignment costs that is interesting in and of by itself, we provide a novel reduction to the (standard) k-median problem, showing that LP-relative guarantees for k-median translate to guarantees for the ordered k-median problem; this yields a nice and clean (8.5+epsilon)-approximation algorithm for {0,1} weights

Approximation Algorithms for Clustering with Dynamic Points

In many classic clustering problems, we seek to sketch a massive data set of $n$ points in a metric space, by segmenting them into $k$ categories or clusters, each cluster represented concisely by a single point in the metric space. Two notable examples are the $k$-center/$k$-supplier problem and the $k$-median problem. In practical applications of clustering, the data set may evolve over time, reflecting an evolution of the underlying clustering model. In this paper, we initiate the study of a dynamic version of clustering problems that aims to capture these considerations. In this version there are $T$ time steps, and in each time step $t\in\{1,2,\dots,T\}$, the set of clients needed to be clustered may change, and we can move the $k$ facilities between time steps. More specifically, we study two concrete problems in this framework: the Dynamic Ordered $k$-Median and the Dynamic $k$-Supplier problem. We first consider the Dynamic Ordered $k$-Median problem, where the objective is to minimize the weighted sum of ordered distances over all time steps, plus the total cost of moving the facilities between time steps. We present one constant-factor approximation algorithm for $T=2$ and another approximation algorithm for fixed $T \geq 3$. Then we consider the Dynamic $k$-Supplier problem, where the objective is to minimize the maximum distance from any client to its facility, subject to the constraint that between time steps the maximum distance moved by any facility is no more than a given threshold. When the number of time steps $T$ is 2, we present a simple constant factor approximation algorithm and a bi-criteria constant factor approximation algorithm for the outlier version, where some of the clients can be discarded. We also show that it is NP-hard to approximate the problem with any factor for $T \geq 3$.Comment: To be published in the Proceedings of the 28th Annual European Symposium on Algorithms (ESA 2020