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 $\epsilon > 0$

Experimental demonstration of near-infrared negative-index metamaterials

Metal-based negative refractive-index materials have been extensively studied in the microwave region. However, negative-index metamaterials have not been realized at near-IR or visible frequencies due to difficulties of fabrication and to the generally poor optical properties of metals at these wavelengths. In this Letter, we report the first fabrication and experimental verification of a transversely structured metal-dielectricmetal multilayer exhibiting a negative refractive index around 2 mu m. Both the amplitude and the phase of the transmission and reflection were measured experimentally, and are in good agreement with a rigorous coupled wave analysis

Capacitated Center Problems with Two-Sided Bounds and Outliers

In recent years, the capacitated center problems have attracted a lot of research interest. Given a set of vertices $V$, we want to find a subset of vertices $S$, called centers, such that the maximum cluster radius is minimized. Moreover, each center in $S$ should satisfy some capacity constraint, which could be an upper or lower bound on the number of vertices it can serve. Capacitated $k$-center problems with one-sided bounds (upper or lower) have been well studied in previous work, and a constant factor approximation was obtained. We are the first to study the capacitated center problem with both capacity lower and upper bounds (with or without outliers). We assume each vertex has a uniform lower bound and a non-uniform upper bound. For the case of opening exactly $k$ centers, we note that a generalization of a recent LP approach can achieve constant factor approximation algorithms for our problems. Our main contribution is a simple combinatorial algorithm for the case where there is no cardinality constraint on the number of open centers. Our combinatorial algorithm is simpler and achieves better constant approximation factor compared to the LP approach

Incremental Medians via Online Bidding

In the k-median problem we are given sets of facilities and customers, and distances between them. For a given set F of facilities, the cost of serving a customer u is the minimum distance between u and a facility in F. The goal is to find a set F of k facilities that minimizes the sum, over all customers, of their service costs. Following Mettu and Plaxton, we study the incremental medians problem, where k is not known in advance, and the algorithm produces a nested sequence of facility sets where the kth set has size k. The algorithm is c-cost-competitive if the cost of each set is at most c times the cost of the optimum set of size k. We give improved incremental algorithms for the metric version: an 8-cost-competitive deterministic algorithm, a 2e ~ 5.44-cost-competitive randomized algorithm, a (24+epsilon)-cost-competitive, poly-time deterministic algorithm, and a (6e+epsilon ~ .31)-cost-competitive, poly-time randomized algorithm. The algorithm is s-size-competitive if the cost of the kth set is at most the minimum cost of any set of size k, and has size at most s k. The optimal size-competitive ratios for this problem are 4 (deterministic) and e (randomized). We present the first poly-time O(log m)-size-approximation algorithm for the offline problem and first poly-time O(log m)-size-competitive algorithm for the incremental problem. Our proofs reduce incremental medians to the following online bidding problem: faced with an unknown threshold T, an algorithm submits "bids" until it submits a bid that is at least the threshold. It pays the sum of all its bids. We prove that folklore algorithms for online bidding are optimally competitive.Comment: conference version appeared in LATIN 2006 as "Oblivious Medians via Online Bidding

Maximum gradient embeddings and monotone clustering

Let (X,d_X) be an n-point metric space. We show that there exists a distribution D over non-contractive embeddings into trees f:X-->T such that for every x in X, the expectation with respect to D of the maximum over y in X of the ratio d_T(f(x),f(y)) / d_X(x,y) is at most C (log n)^2, where C is a universal constant. Conversely we show that the above quadratic dependence on log n cannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location.Comment: 25 pages, 2 figures. Final version, minor revision of the previous one. To appear in "Combinatorica

A Clinical Practice Guideline for the Management of Patients With Acute Spinal Cord Injury: Recommendations on Hemodynamic Management

STUDY DESIGN Clinical practice guideline development following the GRADE process. OBJECTIVES Hemodynamic management is one of the only available treatment options that likely improves neurologic outcomes in patients with acute traumatic spinal cord injury (SCI). Augmenting mean arterial pressure (MAP) aims to improve blood perfusion and oxygen delivery to the injured spinal cord in order to minimize secondary ischemic damage to neural tissue. The objective of this guideline was to update the 2013 AANS/CNS recommendations on the hemodynamic management of patients with acute traumatic SCI, acknowledging that much has been published in this area since its publication. Specifically, we sought to make recommendations on 1. The range of mean arterial pressure (MAP) to be maintained by identifying an upper and lower MAP limit; 2. The duration of such MAP augmentation; and 3. The choice of vasopressor. Additionally, we sought to make a recommendation on spinal cord perfusion pressure (SCPP) targets. METHODS A multidisciplinary guideline development group (GDG) was formed that included health care professionals from a wide range of clinical specialities, patient advocates, and individuals living with SCI. The GDG reviewed the 2013 AANS/CNS guidelines and voted on whether each recommendation should be endorsed or updated. A systematic review of the literature, following PRISMA standards and registered in PROSPERO, was conducted to inform the guideline development process and address the following key questions: (i) what are the effects of goal-directed interventions to optimize spinal cord perfusion on extent of neurological recovery and rates of adverse events at any time point of follow-up? and (ii) what are the effects of particular monitoring techniques, perfusion ranges, pharmacological agents, and durations of treatment on extent of neurological recovery and rates of adverse events at any time point of follow-up? The GDG combined the information from this systematic review with their clinical expertise in order to develop recommendations on a MAP target range (specifically an upper and lower limit to target), the optimal duration for MAP augmentation, and the use of vasopressors or inotropes. Using methods outlined by the GRADE working group, recommendations were formulated that considered the balance of benefits and harms, financial impact, acceptability, feasibility and patient preferences. RESULTS The GDG suggested that MAP should be augmented to at least 75-80 mmHg as the "lower limit," but not actively augmented beyond an "upper limit" of 90-95 mmHg in order to optimize spinal cord perfusion in acute traumatic SCI. The quality of the evidence around the "target MAP" was very low, and thus the strength of this recommendation is weak. For duration of hemodynamic management, the GDG "suggested" that MAP be augmented for a duration of 3-7 days. Again, the quality of the evidence around the duration of MAP support was very low, and thus the strength of this recommendation is also weak. The GDG felt that a recommendation on the choice of vasopressor or the use of SCPP targets was not warranted, given the dearth of available evidence. CONCLUSION We provide new recommendations for blood pressure management after acute SCI that acknowledge the limitations of the current evidence on the relationship between MAP and neurologic recovery. It was felt that the low quality of existing evidence and uncertainty around the relationship between MAP and neurologic recovery justified a greater range of MAP to target, and for a broader range of days post-injury than recommended in previous guidelines. While important knowledge gaps still remain regarding hemodynamic management, these recommendations represent current perspectives on the role of MAP augmentation for acute SCI

Dynamic temporary blood facility location-allocation during and post-disaster periods

The key objective of this study is to develop a tool (hybridization or integration of different techniques) for locating the temporary blood banks during and post-disaster conditions that could serve the hospitals with minimum response time. We have used temporary blood centers, which must be located in such a way that it is able to serve the demand of hospitals in nearby region within a shorter duration. We are locating the temporary blood centres for which we are minimizing the maximum distance with hospitals. We have used Tabu search heuristic method to calculate the optimal number of temporary blood centres considering cost components. In addition, we employ Bayesian belief network to prioritize the factors for locating the temporary blood facilities. Workability of our model and methodology is illustrated using a case study including blood centres and hospitals surrounding Jamshedpur city. Our results shows that at-least 6 temporary blood facilities are required to satisfy the demand of blood during and post-disaster periods in Jamshedpur. The results also show that that past disaster conditions, response time and convenience for access are the most important factors for locating the temporary blood facilities during and post-disaster periods

Optimal methods for coordinated en-route web caching for tree networks

Web caching is an important technology for improving the scalability of Web services. One of the key problems in coordinated enroute Web caching is to compute the locations for storing copies of an object among the enroute caches so that some specified objectives are achieved. In this article, we address this problem for tree networks, and formulate it as a maximization problem. We consider this problem for both unconstrained and constrained cases. The constrained case includes constraints on the cost gain per node and on the number of object copies to be placed. We present dynamic programming-based solutions to this problem for different cases and theoretically show that the solutions are either optimal or convergent to optimal solutions. We derive efficient algorithms that produce these solutions. Based on our mathematical model, we also present a solution to coordinated enroute Web caching for autonomous systems as a natural extension of the solution for tree networks. We implement our algorithms and evaluate our model on different performance metrics through extensive simulation experiments. The implementation results show that our methods outperform the existing algorithms of either coordinated enroute Web caching for linear topology or object placement (replacement) at individual nodes only.Keqiu Li, Hong Shen, Francis Y. L. Chin, Si Qing Zhen

A Clinical Practice Guideline for the Management of Patients With Acute Spinal Cord Injury: Recommendations on Hemodynamic Management

STUDY DESIGN: Clinical practice guideline development following the GRADE process. OBJECTIVES: Hemodynamic management is one of the only available treatment options that likely improves neurologic outcomes in patients with acute traumatic spinal cord injury (SCI). Augmenting mean arterial pressure (MAP) aims to improve blood perfusion and oxygen delivery to the injured spinal cord in order to minimize secondary ischemic damage to neural tissue. The objective of this guideline was to update the 2013 AANS/CNS recommendations on the hemodynamic management of patients with acute traumatic SCI, acknowledging that much has been published in this area since its publication. Specifically, we sought to make recommendations on 1. The range of mean arterial pressure (MAP) to be maintained by identifying an upper and lower MAP limit; 2. The duration of such MAP augmentation; and 3. The choice of vasopressor. Additionally, we sought to make a recommendation on spinal cord perfusion pressure (SCPP) targets. METHODS: A multidisciplinary guideline development group (GDG) was formed that included health care professionals from a wide range of clinical specialities, patient advocates, and individuals living with SCI. The GDG reviewed the 2013 AANS/CNS guidelines and voted on whether each recommendation should be endorsed or updated. A systematic review of the literature, following PRISMA standards and registered in PROSPERO, was conducted to inform the guideline development process and address the following key questions: (i) what are the effects of goal-directed interventions to optimize spinal cord perfusion on extent of neurological recovery and rates of adverse events at any time point of follow-up? and (ii) what are the effects of particular monitoring techniques, perfusion ranges, pharmacological agents, and durations of treatment on extent of neurological recovery and rates of adverse events at any time point of follow-up? The GDG combined the information from this systematic review with their clinical expertise in order to develop recommendations on a MAP target range (specifically an upper and lower limit to target), the optimal duration for MAP augmentation, and the use of vasopressors or inotropes. Using methods outlined by the GRADE working group, recommendations were formulated that considered the balance of benefits and harms, financial impact, acceptability, feasibility and patient preferences. RESULTS: The GDG suggested that MAP should be augmented to at least 75-80 mmHg as the lower limit, but not actively augmented beyond an upper limit of 90-95 mmHg in order to optimize spinal cord perfusion in acute traumatic SCI. The quality of the evidence around the target MAP was very low, and thus the strength of this recommendation is weak. For duration of hemodynamic management, the GDG suggested that MAP be augmented for a duration of 3-7 days. Again, the quality of the evidence around the duration of MAP support was very low, and thus the strength of this recommendation is also weak. The GDG felt that a recommendation on the choice of vasopressor or the use of SCPP targets was not warranted, given the dearth of available evidence. CONCLUSION: We provide new recommendations for blood pressure management after acute SCI that acknowledge the limitations of the current evidence on the relationship between MAP and neurologic recovery. It was felt that the low quality of existing evidence and uncertainty around the relationship between MAP and neurologic recovery justified a greater range of MAP to target, and for a broader range of days post-injury than recommended in previous guidelines. While important knowledge gaps still remain regarding hemodynamic management, these recommendations represent current perspectives on the role of MAP augmentation for acute SCI