Scalable Facility Location for Massive Graphs on Pregel-like Systems

We propose a new scalable algorithm for facility location. Facility location is a classic problem, where the goal is to select a subset of facilities to open, from a set of candidate facilities F , in order to serve a set of clients C. The objective is to minimize the total cost of opening facilities plus the cost of serving each client from the facility it is assigned to. In this work, we are interested in the graph setting, where the cost of serving a client from a facility is represented by the shortest-path distance on the graph. This setting allows to model natural problems arising in the Web and in social media applications. It also allows to leverage the inherent sparsity of such graphs, as the input is much smaller than the full pairwise distances between all vertices. To obtain truly scalable performance, we design a parallel algorithm that operates on clusters of shared-nothing machines. In particular, we target modern Pregel-like architectures, and we implement our algorithm on Apache Giraph. Our solution makes use of a recent result to build sketches for massive graphs, and of a fast parallel algorithm to find maximal independent sets, as building blocks. In so doing, we show how these problems can be solved on a Pregel-like architecture, and we investigate the properties of these algorithms. Extensive experimental results show that our algorithm scales gracefully to graphs with billions of edges, while obtaining values of the objective function that are competitive with a state-of-the-art sequential algorithm

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 $\epsilon > 0$

Tight Analysis of a Multiple-Swap Heuristic for Budgeted Red-Blue Median

Budgeted Red-Blue Median is a generalization of classic $k$-Median in that there are two sets of facilities, say $\mathcal{R}$ and $\mathcal{B}$, that can be used to serve clients located in some metric space. The goal is to open $k_r$ facilities in $\mathcal{R}$ and $k_b$ facilities in $\mathcal{B}$ for some given bounds $k_r, k_b$ and connect each client to their nearest open facility in a way that minimizes the total connection cost. We extend work by Hajiaghayi, Khandekar, and Kortsarz [2012] and show that a multiple-swap local search heuristic can be used to obtain a $(5+\epsilon)$-approximation for Budgeted Red-Blue Median for any constant $\epsilon > 0$. This is an improvement over their single swap analysis and beats the previous best approximation guarantee of 8 by Swamy [2014]. We also present a matching lower bound showing that for every $p \geq 1$, there are instances of Budgeted Red-Blue Median with local optimum solutions for the $p$-swap heuristic whose cost is $5 + \Omega\left(\frac{1}{p}\right)$ times the optimum solution cost. Thus, our analysis is tight up to the lower order terms. In particular, for any $\epsilon > 0$ we show the single-swap heuristic admits local optima whose cost can be as bad as $7-\epsilon$ times the optimum solution cost

The Unreasonable Success of Local Search: Geometric Optimization

What is the effectiveness of local search algorithms for geometric problems in the plane? We prove that local search with neighborhoods of magnitude $1/\epsilon^c$ is an approximation scheme for the following problems in the Euclidian plane: TSP with random inputs, Steiner tree with random inputs, facility location (with worst case inputs), and bicriteria $k$-median (also with worst case inputs). The randomness assumption is necessary for TSP

Sherali-Adams gaps, flow-cover inequalities and generalized configurations for capacity-constrained Facility Location

Metric facility location is a well-studied problem for which linear programming methods have been used with great success in deriving approximation algorithms. The capacity-constrained generalizations, such as capacitated facility location (CFL) and lower-bounded facility location (LBFL), have proved notorious as far as LP-based approximation is concerned: while there are local-search-based constant-factor approximations, there is no known linear relaxation with constant integrality gap. According to Williamson and Shmoys devising a relaxation-based approximation for \cfl\ is among the top 10 open problems in approximation algorithms. This paper advances significantly the state-of-the-art on the effectiveness of linear programming for capacity-constrained facility location through a host of impossibility results for both CFL and LBFL. We show that the relaxations obtained from the natural LP at $\Omega(n)$ levels of the Sherali-Adams hierarchy have an unbounded gap, partially answering an open question of \cite{LiS13, AnBS13}. Here, $n$ denotes the number of facilities in the instance. Building on the ideas for this result, we prove that the standard CFL relaxation enriched with the generalized flow-cover valid inequalities \cite{AardalPW95} has also an unbounded gap. This disproves a long-standing conjecture of \cite{LeviSS12}. We finally introduce the family of proper relaxations which generalizes to its logical extreme the classic star relaxation and captures general configuration-style LPs. We characterize the behavior of proper relaxations for CFL and LBFL through a sharp threshold phenomenon.Comment: arXiv admin note: substantial text overlap with arXiv:1305.599

LP-Based Algorithms for Capacitated Facility Location

Linear programming has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of algorithmic methodologies, such as LP-rounding and primal-dual method, have been applied to and evolved from algorithms for this problem. Unfortunately, this collection of powerful algorithmic techniques had not yet been applicable to the more general capacitated facility location problem. In fact, all of the known algorithms with good performance guarantees were based on a single technique, local search, and no linear programming relaxation was known to efficiently approximate the problem. In this paper, we present a linear programming relaxation with constant integrality gap for capacitated facility location. We demonstrate that the fundamental theories of multi-commodity flows and matchings provide key insights that lead to the strong relaxation. Our algorithmic proof of integrality gap is obtained by finally accessing the rich toolbox of LP-based methodologies: we present a constant factor approximation algorithm based on LP-rounding.Comment: 25 pages, 6 figures; minor revision

Strengthening Pharmacovigilance System to Capture Safety Data from HIV Clients on ART in Tanzania: Identification of Gaps in Safety Reporting System

In Tanzania, pharmacovigilance system is implemented by Tanzania Food and Drugs Authority (TFDA) that monitors drug use countrywide. TFDA is the main national custodian for recording, analyzing and disseminating safety information that is generated through conventional health care facilities. Since the introduction of Care and Treatment Centre (CTC) in the health care system, little has been achieved on translating safety information from these facilities to the TFDA. Since the inception of national pharmacovigilance framework in 2003 there has been no systematic operational research to map the gaps in the existing pharmacovigilance system. Furthermore, it is not clear if there is adequate training and supervision. It is, therefore, important to strengthen antiretroviral therapy (ART) related adverse drug reactions (ADRs) reporting by mapping gaps in implementation of pharmacovigilance (PV) system. Information obtained will assist in addressing training needs to ensure effective reporting of ADRs through coordinated approach involving TFDA and National AIDS Control Program (NACP) in Tanzania. A cross-sectional study was conducted in four regions (Tanga, Singida, Dodoma and Mtwara) in two PV zones. Qualitative and quantitative data collection techniques with triangulation design were used. These included; desk document review of PV recording and reporting of drug safety information; in-depth interviews with various implementation stakeholders, exit interviews with patients, in-interviews with care takers and community based organizations (CBOs) involved in the provision of care and treatment of HIV/AIDS. A total of 801 respondents participated in the quantitative data component which included; 545 exit interviews to CTC clients, 177 health service providers, 62 in-depth interviews to CTC in-charges and 17 regional and district pharmacists. Ownership of these CTCs included 83.9% government, 12.9% faith based organizations and 3.2% co-owned by the government and faith based organizations. High proportions (97.2%) of the CTC health care providers had wide knowledge on ART related ADRs. However, more than half (53.4%) of the CTC service providers had not attended any training on ART related ADRs. Among the service providers, majority (67.8%) mentioned there was no guideline in place for reporting ART related ADRs. Only, 32.1% of health care providers indicated to be aware of the tool used for collection of ART related ADRs events. Of those, 37.5% mentioned that the forms were mainly obtained from district or regional pharmacists. The ADR reports were submitted to district and regional pharmacists 48.3%, TFDA 7.0%, and NACP 7.0%. Of those who indicated to have filled and submitted ADR form, only 7.4% received feedback. The proportion of ART clients who provided information was significantly different between urban and rural in Dodoma region (p=0.002). There was variation in proportions of ART clients who had mentioned seen/heard of ART related ADR by regions and difference was significant between rural and urban for all regions except Tanga (p<0.05). Majority (47.9%) of the ART clients reported ART related ADRs to the health provider for duration ranging from 3-7 days. The qualitative results revealed that that most of the guidelines from TFDA were not known and unavailable according to most of the respondents at national level (NACP), regional, district, and at health facility level. It was surprising that one of the district pharmacists interviewed was unaware of existence of guidelines in place for ADR and PV for use in the districts. It was also found that Sometimes even when available at health facilities, there was inadequate knowledge on how to fill the ADR forms according to Key Informant at national level. Moreover, several health workers admitted that that they were not reporting ADR due to a lack of forms according to some CTC in-charges interviewed. This study has shown that despite the established PV system in Tanzania, the frequency of reporting of ART related ADRs to TFDA is low. This is due to inadequate training of health care providers on ADR reporting, shortage of staff, unavailability of TFDA ADR reporting forms and lack of regular supportive supervision. Based on these results therefore we recommend TFDA should ensure that ADR reporting forms as well as guidelines are adequately supplied and utilized at CTC level NACP should ensure sharing of safety information with TFDA and recommend dedicated focal person liable for documenting and reporting ART related ADRs recorded in CTC II patient file. Regular training, supportive supervision and feedback on ART related ADR reporting system for health care providers is needed. The financial support was provided by the Global Fund Round 8. The total budget for the project was Tsh. 69,993,000/-

A simulated annealing algorithm for router nodes placement problem in Wireless Mesh Networks

Mesh router nodes placement is a central problem in Wireless Mesh Networks (WMNs). An efficient placement of mesh router nodes is indispensable for achieving network performance in terms of both network connectivity and user coverage. Unfortunately the problem is computationally hard to solve to optimality even for small deployment areas and a small number of mesh router nodes. As WMNs are becoming an important networking infrastructure for providing cost-efficient broadband wireless connectivity, researchers are paying attention to the resolution of the mesh router placement problem through heuristic approaches in order to achieve near optimal, yet high quality solutions in reasonable time. In this work we propose and evaluate a simulated annealing (SA) approach to placement of mesh router nodes in WMNs. The optimization model uses two maximization objectives, namely, the size of the giant component in the network and user coverage. Both objectives are important to deployment of WMNs; the former is crucial to achieve network connectivity while the later is an indicator of the QoS in WMNs. The SA approach distinguishes for its simplicity yet its policy of neighborhood exploration allows to reach promising areas of the solution space where quality solutions could be found. We have experimentally evaluated the SA algorithm through a benchmark of generated instances, varying from small to large size, and capturing different characteristics of WMNs such as topological placements of mesh clients. The experimental results showed the efficiency of the annealing approach for the placement of mesh router nodes in WMNs.Peer ReviewedPostprint (author's final draft