We consider the online facility assignment problem, with a set of facilities F of equal capacity l in metric space and customers arriving one by one in an online manner. We must assign customer ci to facility fj before the next customer ci+1 arrives. The cost of this assignment is the distance between ci and fj. The total number of customers is at most Fl and each customer must be assigned to a facility. The objective is to minimize the sum of all assignment costs. We first consider the case where facilities are placed on a line so that the distance between adjacent facilities is the same and customers appear anywhere on the line. We describe a greedy algorithm with competitive ratio 4F and another one with competitive ratio F. Finally, we consider a variant in which the facilities are placed on the vertices of a graph and two algorithms in that setting.
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This content will become publicly available on July 7, 2024
Which Lp norm is the fairest? Approximations for fair facility location across all "p"
Given a set of facilities and clients, and costs to open facilities, the classic facility location problem seeks to
open a set of facilities and assign each client to one open facility to minimize the cost of opening the chosen
facilities and the total distance of the clients to their assigned open facilities. Such an objective may induce an
unequal cost over certain socioeconomic groups of clients (i.e., total distance traveled by clients in such a
group). This is important when planning the location of socially relevant facilities such as emergency rooms.
In this work, we consider a fair version of the problem where we are given π clients groups that partition
the set of clients, and the distance of a given group is defined as the average distance of clients in the group
to their respective open facilities. The objective is to minimize the Minkowski πnorm of vector of group
distances, to penalize high access costs to open facilities across π groups of clients. This generalizes classic
facility location (π = 1) and the minimization of the maximum group distance (π = β). However, in practice,
fairness criteria may not be explicit or even known to a decision maker, and it is often unclear how to select a
specific "π" to model the cost of unfairness. To get around this, we study the notion of solution portfolios where
for a fixed problem instance, we seek a small portfolio of solutions such that for any Minkowski norm π, one
of these solutions is an π(1)approximation. Using the geometric relationship between various πnorms, we
show the existence of a portfolio of cardinality π(log π), and a lower bound of (\sqrt{log r}).
There may not be common structure across different solutions in this portfolio, which can make planning
difficult if the notion of fairness changes over time or if the budget to open facilities is disbursed over time. For
example, small changes in π could lead to a completely different set of open facilities in the portfolio. Inspired
by this, we introduce the notion of refinement, which is a family of solutions for each πnorm satisfying a
combinatorial property. This property requires that (1) the set of facilities open for a higher πnorm must be
a subset of the facilities open for a lower πnorm, and (2) all clients assigned to an open facility for a lower
πnorm must be assigned to the same open facility for any higher πnorm. A refinement is πΌapproximate if
the solution for each πnorm problem is an πΌapproximation for it. We show that it is sufficient to consider
only π(log π) norms instead of all πnorms, π β [1, β] to construct refinements. A natural greedy algorithm
for the problem gives a poly(π)approximate refinement, which we improve to poly(r^1/\sqrt{log π})approximate
using a recursive algorithm. We improve this ratio to π(log π) for the special case of tree metric for uniform
facility open cost. Our recursive algorithm extends to other settings, including to a hierarchical facility location
problem that models facility location problems at several levels, such as public works departments and schools.
A full version of this paper can be found at https://arxiv.org/abs/2211.14873.
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 NSFPAR ID:
 10473304
 Publisher / Repository:
 ACM
 Date Published:
 Journal Name:
 EC '23: Proceedings of the 24th ACM Conference on Economics and Computation
 Page Range / eLocation ID:
 817 to 817
 Format(s):
 Medium: X
 Location:
 London United Kingdom
 Sponsoring Org:
 National Science Foundation
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Random dimensionality reduction is a versatile tool for speeding up algorithms for highdimensional problems. We study its application to two clustering problems: the facility location problem, and the singlelinkage hierarchical clustering problem, which is equivalent to computing the minimum spanning tree. We show that if we project the input pointset π onto a random π=π(ππ)dimensional subspace (where ππ is the doubling dimension of π), then the optimum facility location cost in the projected space approximates the original cost up to a constant factor. We show an analogous statement for minimum spanning tree, but with the dimension π having an extra loglogπ term and the approximation factor being arbitrarily close to 1. Furthermore, we extend these results to approximating solutions instead of just their costs. Lastly, we provide experimental results to validate the quality of solutions and the speedup due to the dimensionality reduction. Unlike several previous papers studying this approach in the context of πmeans and πmedians, our dimension bound does not depend on the number of clusters but only on the intrinsic dimensionality of π.more » « less

Obeid, I. (Ed.)The Neural Engineering Data Consortium (NEDC) is developing the Temple University Digital Pathology Corpus (TUDP), an open source database of highresolution images from scanned pathology samples [1], as part of its National Science Foundationfunded Major Research Instrumentation grant titled βMRI: High Performance Digital Pathology Using Big Data and Machine Learningβ [2]. The longterm goal of this project is to release one million images. We have currently scanned over 100,000 images and are in the process of annotating breast tissue data for our first official corpus release, v1.0.0. This release contains 3,505 annotated images of breast tissue including 74 patients with cancerous diagnoses (out of a total of 296 patients). In this poster, we will present an analysis of this corpus and discuss the challenges we have faced in efficiently producing high quality annotations of breast tissue. It is well known that state of the art algorithms in machine learning require vast amounts of data. Fields such as speech recognition [3], image recognition [4] and text processing [5] are able to deliver impressive performance with complex deep learning models because they have developed large corpora to support training of extremely highdimensional models (e.g., billions of parameters). Other fields that do not have access to such data resources must rely on techniques in which existing models can be adapted to new datasets [6]. A preliminary version of this breast corpus release was tested in a pilot study using a baseline machine learning system, ResNet18 [7], that leverages several opensource Python tools. The pilot corpus was divided into three sets: train, development, and evaluation. Portions of these slides were manually annotated [1] using the nine labels in Table 1 [8] to identify five to ten examples of pathological features on each slide. Not every pathological feature is annotated, meaning excluded areas can include focuses particular to these labels that are not used for training. A summary of the number of patches within each label is given in Table 2. To maintain a balanced training set, 1,000 patches of each label were used to train the machine learning model. Throughout all sets, only annotated patches were involved in model development. The performance of this model in identifying all the patches in the evaluation set can be seen in the confusion matrix of classification accuracy in Table 3. The highest performing labels were background, 97% correct identification, and artifact, 76% correct identification. A correlation exists between labels with more than 6,000 development patches and accurate performance on the evaluation set. Additionally, these results indicated a need to further refine the annotation of invasive ductal carcinoma (βindcβ), inflammation (βinflβ), nonneoplastic features (βnneoβ), normal (βnormβ) and suspicious (βsuspβ). This pilot experiment motivated changes to the corpus that will be discussed in detail in this poster presentation. To increase the accuracy of the machine learning model, we modified how we addressed underperforming labels. One common source of error arose with how nonbackground labels were converted into patches. Large areas of background within other labels were isolated within a patch resulting in connective tissue misrepresenting a nonbackground label. In response, the annotation overlay margins were revised to exclude benign connective tissue in nonbackground labels. Corresponding patient reports and supporting immunohistochemical stains further guided annotation reviews. The microscopic diagnoses given by the primary pathologist in these reports detail the pathological findings within each tissue site, but not within each specific slide. The microscopic diagnoses informed revisions specifically targeting annotated regions classified as cancerous, ensuring that the labels βindcβ and βdcisβ were used only in situations where a micropathologist diagnosed it as such. Further differentiation of cancerous and precancerous labels, as well as the location of their focus on a slide, could be accomplished with supplemental immunohistochemically (IHC) stained slides. When distinguishing whether a focus is a nonneoplastic feature versus a cancerous growth, pathologists employ antigen targeting stains to the tissue in question to confirm the diagnosis. For example, a nonneoplastic feature of usual ductal hyperplasia will display diffuse staining for cytokeratin 5 (CK5) and no diffuse staining for estrogen receptor (ER), while a cancerous growth of ductal carcinoma in situ will have negative or focally positive staining for CK5 and diffuse staining for ER [9]. Many tissue samples contain cancerous and noncancerous features with morphological overlaps that cause variability between annotators. The informative fields IHC slides provide could play an integral role in machine model pathology diagnostics. Following the revisions made on all the annotations, a second experiment was run using ResNet18. Compared to the pilot study, an increase of model prediction accuracy was seen for the labels indc, infl, nneo, norm, and null. This increase is correlated with an increase in annotated area and annotation accuracy. Model performance in identifying the suspicious label decreased by 25% due to the decrease of 57% in the total annotated area described by this label. A summary of the model performance is given in Table 4, which shows the new prediction accuracy and the absolute change in error rate compared to Table 3. The breast tissue subset we are developing includes 3,505 annotated breast pathology slides from 296 patients. The average size of a scanned SVS file is 363 MB. The annotations are stored in an XML format. A CSV version of the annotation file is also available which provides a flat, or simple, annotation that is easy for machine learning researchers to access and interface to their systems. Each patient is identified by an anonymized medical reference number. Within each patientβs directory, one or more sessions are identified, also anonymized to the first of the month in which the sample was taken. These sessions are broken into groupings of tissue taken on that date (in this case, breast tissue). A deidentified patient report stored as a flat text file is also available. Within these slides there are a total of 16,971 total annotated regions with an average of 4.84 annotations per slide. Among those annotations, 8,035 are noncancerous (normal, background, null, and artifact,) 6,222 are carcinogenic signs (inflammation, nonneoplastic and suspicious,) and 2,714 are cancerous labels (ductal carcinoma in situ and invasive ductal carcinoma in situ.) The individual patients are split up into three sets: train, development, and evaluation. Of the 74 cancerous patients, 20 were allotted for both the development and evaluation sets, while the remain 34 were allotted for train. The remaining 222 patients were split up to preserve the overall distribution of labels within the corpus. This was done in hope of creating control sets for comparable studies. Overall, the development and evaluation sets each have 80 patients, while the training set has 136 patients. In a related component of this project, slides from the Fox Chase Cancer Center (FCCC) Biosample Repository (https://www.foxchase.org/research/facilities/geneticresearchfacilities/biosamplerepository facility) are being digitized in addition to slides provided by Temple University Hospital. This data includes 18 different types of tissue including approximately 38.5% urinary tissue and 16.5% gynecological tissue. These slides and the metadata provided with them are already anonymized and include diagnoses in a spreadsheet with sample and patient ID. We plan to release over 13,000 unannotated slides from the FCCC Corpus simultaneously with v1.0.0 of TUDP. Details of this release will also be discussed in this poster. Few digitally annotated databases of pathology samples like TUDP exist due to the extensive data collection and processing required. The breast corpus subset should be released by November 2021. By December 2021 we should also release the unannotated FCCC data. We are currently annotating urinary tract data as well. We expect to release about 5,600 processed TUH slides in this subset. We have an additional 53,000 unprocessed TUH slides digitized. Corpora of this size will stimulate the development of a new generation of deep learning technology. In clinical settings where resources are limited, an assistive diagnoses model could support pathologistsβ workload and even help prioritize suspected cancerous cases. ACKNOWLEDGMENTS This material is supported by the National Science Foundation under grants nos. CNS1726188 and 1925494. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. REFERENCES [1] N. Shawki et al., βThe Temple University Digital Pathology Corpus,β in Signal Processing in Medicine and Biology: Emerging Trends in Research and Applications, 1st ed., I. Obeid, I. Selesnick, and J. Picone, Eds. New York City, New York, USA: Springer, 2020, pp. 67 104. https://www.springer.com/gp/book/9783030368432. [2] J. Picone, T. Farkas, I. Obeid, and Y. Persidsky, βMRI: High Performance Digital Pathology Using Big Data and Machine Learning.β Major Research Instrumentation (MRI), Division of Computer and Network Systems, Award No. 1726188, January 1, 2018 β December 31, 2021. https://www. isip.piconepress.com/projects/nsf_dpath/. [3] A. Gulati et al., βConformer: Convolutionaugmented Transformer for Speech Recognition,β in Proceedings of the Annual Conference of the International Speech Communication Association (INTERSPEECH), 2020, pp. 50365040. https://doi.org/10.21437/interspeech.20203015. [4] C.J. Wu et al., βMachine Learning at Facebook: Understanding Inference at the Edge,β in Proceedings of the IEEE International Symposium on High Performance Computer Architecture (HPCA), 2019, pp. 331β344. https://ieeexplore.ieee.org/document/8675201. [5] I. Caswell and B. Liang, βRecent Advances in Google Translate,β Google AI Blog: The latest from Google Research, 2020. [Online]. Available: https://ai.googleblog.com/2020/06/recentadvancesingoogletranslate.html. [Accessed: 01Aug2021]. [6] V. Khalkhali, N. Shawki, V. Shah, M. Golmohammadi, I. Obeid, and J. Picone, βLow Latency RealTime Seizure Detection Using Transfer Deep Learning,β in Proceedings of the IEEE Signal Processing in Medicine and Biology Symposium (SPMB), 2021, pp. 1 7. https://www.isip. piconepress.com/publications/conference_proceedings/2021/ieee_spmb/eeg_transfer_learning/. [7] J. Picone, T. Farkas, I. Obeid, and Y. Persidsky, βMRI: High Performance Digital Pathology Using Big Data and Machine Learning,β Philadelphia, Pennsylvania, USA, 2020. https://www.isip.piconepress.com/publications/reports/2020/nsf/mri_dpath/. [8] I. Hunt, S. Husain, J. Simons, I. Obeid, and J. Picone, βRecent Advances in the Temple University Digital Pathology Corpus,β in Proceedings of the IEEE Signal Processing in Medicine and Biology Symposium (SPMB), 2019, pp. 1β4. https://ieeexplore.ieee.org/document/9037859. [9] A. P. Martinez, C. Cohen, K. Z. Hanley, and X. (Bill) Li, βEstrogen Receptor and Cytokeratin 5 Are Reliable Markers to Separate Usual Ductal Hyperplasia From Atypical Ductal Hyperplasia and LowGrade Ductal Carcinoma In Situ,β Arch. Pathol. Lab. Med., vol. 140, no. 7, pp. 686β689, Apr. 2016. https://doi.org/10.5858/arpa.20150238OA.more » « less

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