Centrality of Trees for Capacitated k-Center

Abstract

We consider the capacitated k-center problem. In this problem we are given a finite set of locations in a metric space and each location has an associated non-negative integer capacity. The goal is to choose (open) k locations (called centers) and assign each location to an open center to minimize the maximum, over all locations, of the distance of the location to its assigned center. The number of locations assigned to a center cannot exceed the center's capacity. The uncapacitated k-center problem has a simple tight 2-approximation from the 80's. In contrast, the first constant factor approximation for the capacitated problem was obtained only recently by Cygan, Hajiaghayi and Khuller who gave an intricate LP-rounding algorithm that achieves an approximation guarantee in the hundreds. In this paper we give a simple algorithm with a clean analysis and prove an approximation guarantee of 9. It uses the standard LP relaxation and comes close to settling the integrality gap (after necessary preprocessing), which is narrowed down to either 7, 8 or 9. The algorithm proceeds by first reducing to special tree instances, and then uses our best-possible algorithm to solve such instances. Our concept of tree instances is versatile and applies to natural variants of the capacitated k-center problem for which we also obtain improved algorithms. Finally, we give evidence to show that more powerful preprocessing could lead to better algorithms, by giving an approximation algorithm that beats the integrality gap for instances where all non-zero capacities are the same.

Improved Approximation Algorithms for Capacitated Fault-Tolerant k-Center

“…Also, if a vertex u was not reassigned, then the distance to its center is at most β. Now, using the 6-approximation for the {0, L}-capacitated k-center by An et al [1], we obtain the following.…”

“…This phase is based on the algorithm of An et al [1] for the capacitated (non-fault-tolerant) k-center. The main difference is that we do not allow transfers from or to vertices in the set B.…”

“…Step The result is a distance-1 transfer y (1) . The first vertex to have its fractional opening transferred is m v , so that, at the end of this step,…”

“…In the second step, we have an integral T -restricted distance-2 transfer of y (1) . Once again, since T does not include vertices of B, we obtain…”

“…It is straightforward to adapt the rounding algorithm for the {0, L}-capacitated k-center by An et al [1], and obtain an integral distance-6 transfer for the solution for this relaxation. The reason that the algorithm obtains a distance-6 transfer for the k-supplier, rather than a distance-5 transfer, is that cluster midpoints are at distance 4 in an instance of the k-supplier, whist midpoints are at distance 3 in an instance of the k-center.…”