On Perturbation Resilience of Non-uniform k-Center

The Non-Uniform k-center (NUkC) problem has recently been formulated by Chakrabarty et al. [ICALP, 2016; ACM Trans Algorithms 16(4):46:1–46:19, 2020] as a generalization of the classical k-center clustering problem. In NUkC, given a set of n points P in a metric space and non-negative numbers $$r_1, r_2, \ldots , r_k$$ r 1 , r 2 , … , r k , the goal is to find the minimum dilation $$\alpha $$ α and to choose k balls centered at the points of P with radius $$\alpha \cdot r_i$$ α · r i for $$1\le i\le k$$ 1 ≤ i ≤ k , such that all points of P are contained in the union of the chosen balls. They showed that the problem is $$\mathsf {NP}$$ NP -hard to approximate within any factor even in tree metrics. On the other hand, they designed a “bi-criteria” constant approximation algorithm that uses a constant times k balls. Surprisingly, no true approximation is known even in the special case when the $$r_i$$ r i ’s belong to a fixed set of size 3. In this paper, we study the NUkC problem under perturbation resilience, which was introduced by Bilu and Linial (Comb Probab Comput 21(5):643–660, 2012). We show that the problem under 2-perturbation resilience is polynomial time solvable when the $$r_i$$ r i ’s belong to a constant-sized set. However, we show that perturbation resilience does not help in the general case. In particular, our findings imply that even with perturbation resilience one cannot hope to find any “good” approximation for the problem.

Embedding of Hypercubes into l-Sibling Trees

The hypercube is one of the most popular interconnection networks due to its structural regularity, potential for parallel computation of various algorithms, and the high degree of fault tolerance. In this paper, we introduce a tree called l-sibling trees and the main results obtained in this paper are: (1) For r ≥ 1, the minimum wirelength of embedding r-dimensional hypercube into r-dimensional l-sibling tree. (2) For r ≥ 1, embedding of r-dimensional extended l-sibling tree into caterpillar with minimum dilation. (3) Based on the proof of (1), we provide an O(r)-linear time algorithm to compute the minimum wirelength of embedding r-dimensional hypercube into r-dimensional l-sibling tree.

APPROXIMATE WEIGHTED FARTHEST NEIGHBORS AND MINIMUM DILATION STARS

We provide an efficient reduction from the problem of querying approximate multiplicatively weighted farthest neighbors in a metric space to the unweighted problem. Combining our techniques with core-sets for approximate unweighted farthest neighbors, we show how to find approximate farthest neighbors that are farther than a factor (1 - ∊) of optimal in time O( log n) per query in D-dimensional Euclidean space for any constants D and ∊. As an application, we find an O(n log n) expected time algorithm for choosing the center of a star topology network connecting a given set of points, so as to approximately minimize the maximum dilation between any pair of points.

COMPUTING GEOMETRIC MINIMUM-DILATION GRAPHS IS NP-HARD

We prove that computing a geometric minimum-dilation graph on a given set of points in the plane, using not more than a given number of edges, is an NP-hard problem, no matter if edge crossings are allowed or forbidden. We also show that the problem remains NP-hard even when a minimum-dilation tour or path is sought; not even an FPTAS exists in this case.