In (k, r)-Center we are given a (possibly edge-weighted) graph and are asked to select at most k vertices (centers), so that all other vertices are at distance at most r from a center. In this paper we provide a number of tight fine-grained bounds on the complexity of this problem with respect to various standard graph parameters. Specifically: • For any r ≥ 1, we show an algorithm that solves ...

We study budget constrained network improvement and degrading problems based on the vertex 1-center problem on graphs: Given a graph with vertex weights and edge lengths the task is to decrease and increase the vertex weights within certain limits such that the optimal 1-center objective value with respect to the new weights is minimized and maximized, respectively. The upgrading (improvement) ...

The k-means problem consists of finding k centers in R that minimize the sum of the squared distances of all points in an input set P from R to their closest respective center. Awasthi et. al. recently showed that there exists a constant ε′ > 0 such that it is NP-hard to approximate the k-means objective within a factor of 1 + ε′. We establish that the constant ε′ is at least 0.0013. For a give...

We consider variants of the metric k-center problem. Imagine that you must choose locations for k firehouses in a city so as to minimize the maximum distance of a house from the nearest firehouse. An instance is specified by a graph with arbitrary nonnegative edge lengths, a set of vertices that can serve as firehouses (i.e., centers) and a set of vertices that represent houses. For general gra...

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, {cij}) with n = |D| points, and a non-increasing weight vector w ∈ Rn+, and the goal is to open k centers and assign each point each point j ∈ D to a center so as to minimize w1 · (largest assignment cost) + w2 · (second-largest assignment cost) + . . . + ...

Facility location problems have always been studied with the assumption that the edge lengths in the network are static and do not change over time. The underlying network could be used to model a city street network for emergency facility location/hospitals, or an electronic network for locating information centers. In any case, it is clear that due to tra c congestion the traversal time on li...

We consider an issue of much current concern: could fairness, an issue that is already difficult toguarantee, worsen when algorithms run much of our lives? We consider this in the context of resource-allocation problems; we show that algorithms can guarantee certain types of fairness in a verifiable way.Our conceptual contribution is a simple approach to fairness in this context...

In this paper we consider several instances of the k-center on a line problem where the goal is, given a set of points S in the plane and a parameter k ≥ 1, to find k disks with centers on a line l such that their union covers S and the maximum radius of the disks is minimized. This problem is a constraint version of the well-known k-center problem in which the centers are constrained to lie in...

We prove that there are one-parameter families of planar differential equations for which the center problem has a trivial solution and on the other hand the cyclicity of the weak focus is arbitrarily high. We illustrate this phenomenon in several examples for which this cyclicity is computed.

We obtain hardness results and approximation algorithms for two related geometric problems involving movement. The first is a constrained variant of the k-center problem, arising from a geometric client-server problem. The second is the problem of moving points towards an independent set.