In this paper, we introduce and study the Non-Uniform k-Center (NUkC) problem. Given a finite metric space (X, d) and a collection of balls of radii {r 1 ≥ • • • ≥ r k }, the NUkC problem is to find a placement of their centers on the metric space and find the minimum dilation α, such that the union of balls of radius α • r i around the ith center covers all the points in X. This problem naturally arises as a min-max vehicle routing problem with fleets of different speeds, or as a wireless router placement problem with routers of different powers/ranges. The NUkC problem generalizes the classic k-center problem when all the k radii are the same (which can be assumed to be 1 after scaling). It also generalizes the k-center with outliers (kCwO for short) problem when there are k balls of radius 1 and balls of radius 0. There are 2-approximation and 3-approximation algorithms known for these problems respectively; the former is best possible unless P=NP and the latter remains unimproved for 15 years. We first observe that no O(1)-approximation is to the optimal dilation is possible unless P=NP, implying that the NUkC problem is more non-trivial than the above two problems. Our main algorithmic result is an (O(1), O(1))-bi-criteria approximation result: we give an O(1)approximation to the optimal dilation, however, we may open Θ(1) centers of each radii. Our techniques also allow us to prove a simple (uni-criteria), optimal 2-approximation to the kCwO problem improving upon the long-standing 3-factor. Our main technical contribution is a connection between the NUkC problem and the so-called firefighter problems on trees which have been studied recently in the TCS community. We show NUkC is as hard as the firefighter problem. While we don't know if the converse is true, we are able to adapt ideas from recent works [4, 1] in non-trivial ways to obtain our constant factor bi-criteria approximation.

Background: Conventional methods of caries removal are commonly associated with pain, fear and discomfort. Chemomechanical methods were introduced to instill a positive dental attitude. Agents like GK-101, Caridex, Carisolv did not prove effective alternatives owing to their high cost, need of special instruments and taste of chlorine. A new chemomechanical agent, Papacarie®, has been introduced to overcome these deficiencies.Objective:This study was aimed to compare the effectiveness and tolerance of Papacárie® with the conventional method. Method:25 children with at least two primary teeth with broad cavitated occlusal or cervical lesion were selected. One carious tooth from each patient was randomly treated with each of Papacarie® and conventional drilling method, one after the other. Time taken for caries excavation, child's pain perception, change in anxiety levels, microbial flora and child's preference of treatment were recorded separately for both the methods. Result:Although the mean time taken for caries removal by the Papacarie® method was slightly longer (P≯0.05) but it led to reduction in pain and anxiety (p<0.05). The viable bacterial counts were significantly reduced by either of the two methods (P <0�0001). More patients preferred Papacarie® over conventional method of treatment (P<0.05). Conclusion: Papacárie® method seems to be a better alternative to conventional method of caries removal.

Objective:To evaluate and compare the efficacy of ferric sulfate, glutaraldehyde, and mineral trioxide aggregate (MTA) as pulpotomy medicaments in primary molars.Materials and Methods:This was a perspective randomized clinical trial. A total of 90 molars from 42 children aged 4–8 years were selected for pulpotomy procedure. Teeth were randomly divided into three equal groups of 30 teeth each. Teeth in Group I were intended to be treated with ferric sulfate, Group II were intended to be treated with buffered glutaraldehyde and Group III with MTA. All the molars were evaluated clinically at 24 h and both clinically and radio graphically at 1, 3, and 6 months. The observations were statistically analyzed using Chi-square test and Fisher's exact test.Results:After 1 month, there was no clinical finding observed in all the three groups. At 3 months postoperative evaluation, 13.3% of teeth in Group I and 12.5% of teeth in Group II had mobility. At 6 months interval, pain and sinus formation each was noted in 9.1% of primary teeth in Group I while periodontal ligament widening was reported in 66.7% of teeth in Group I and 85.7% of teeth in Group II.Conclusion:MTA exhibited overall best results as pulpotomy agent for primary molars followed by 15.5% ferric sulfate, whereas 2% buffered glutaraldehyde was found to be least effective as a pulpotomy agent.

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