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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

We consider the Min-Sum k-Clustering (k-MSC) problem. Given a set of points in a metric which is represented by an edge-weighted graph G = (V, E) and a parameter k, the goal is to partition the points V into k clusters such that the sum of distances between all pairs of the points within the same cluster is minimized.
The k-MSC problem is known to be APX-hard on general metrics. The best known approximation algorithms for the problem obtained by Behsaz, Friggstad, Salavatipour and Sivakumar [Algorithmica 2019] achieve an approximation ratio of O(log |V|) in polynomial time for general metrics and an approximation ratio 2+ε in quasi-polynomial time for metrics with bounded doubling dimension. No approximation schemes for k-MSC (when k is part of the input) is known for any non-trivial metrics prior to our work. In fact, most of the previous works rely on the simple fact that there is a 2-approximate reduction from k-MSC to the balanced k-median problem and design approximation algorithms for the latter to obtain an approximation for k-MSC.
In this paper, we obtain the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem on metrics induced by graphs of bounded treewidth, graphs of bounded highway dimension, graphs of bounded doubling dimensions (including fixed dimensional Euclidean metrics), and planar and minor-free graphs. We bypass the barrier of 2 for k-MSC by introducing a new clustering problem, which we call min-hub clustering, which is a generalization of balanced k-median and is a trade off between center-based clustering problems (such as balanced k-median) and pair-wise clustering (such as Min-Sum k-clustering). We then show how one can find approximation schemes for Min-hub clustering on certain classes of metrics.

Ismail Naderi, Mohsen Rezapour, and Mohammad R. Salavatipour. Approximation Schemes for Min-Sum k-Clustering. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 84:1-84:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{naderi_et_al:LIPIcs.ESA.2023.84, author = {Naderi, Ismail and Rezapour, Mohsen and Salavatipour, Mohammad R.}, title = {{Approximation Schemes for Min-Sum k-Clustering}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {84:1--84:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.84}, URN = {urn:nbn:de:0030-drops-187379}, doi = {10.4230/LIPIcs.ESA.2023.84}, annote = {Keywords: Approximation Algorithms, Clustering, Dynamic Programming} }

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**Published in:** LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

We consider scheduling problems in which jobs need to be processed through a (shared) network of machines. The network is given in the form of a graph the edges of which represent the machines. We are also given a set of jobs, each specified by its processing time and a path in the graph. Every job needs to be processed in the order of edges specified by its path. We assume that jobs can wait between machines and preemption is not allowed; that is, once a job is started being processed on a machine, it must be completed without interruption. Every machine can only process one job at a time.
The makespan of a schedule is the earliest time by which all the jobs have finished processing. The flow time (a.k.a. the completion time) of a job in a schedule is the difference in time between when it finishes processing on its last machine and when the it begins processing on its first machine. The total flow time (or the sum of completion times) is the sum of flow times (or completion times) of all jobs. Our focus is on finding schedules with the minimum sum of completion times or minimum makespan.
In this paper, we develop several algorithms (both approximate and exact) for the problem both on general graphs and when the underlying graph of machines is a tree. Even in the very special case when the underlying network is a simple star, the problem is very interesting as it models a biprocessor scheduling with applications to data migration.

Zachary Friggstad, Arnoosh Golestanian, Kamyar Khodamoradi, Christopher Martin, Mirmahdi Rahgoshay, Mohsen Rezapour, Mohammad R. Salavatipour, and Yifeng Zhang. Scheduling Problems over Network of Machines. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 5:1-5:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{friggstad_et_al:LIPIcs.APPROX-RANDOM.2017.5, author = {Friggstad, Zachary and Golestanian, Arnoosh and Khodamoradi, Kamyar and Martin, Christopher and Rahgoshay, Mirmahdi and Rezapour, Mohsen and Salavatipour, Mohammad R. and Zhang, Yifeng}, title = {{Scheduling Problems over Network of Machines}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {5:1--5:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.5}, URN = {urn:nbn:de:0030-drops-75547}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.5}, annote = {Keywords: approximation algorithms, job-shop scheduling, min-sum edge coloring, minimum latency} }

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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

We consider a lower- and upper-bounded generalization of the classical facility location problem, where each facility has a capacity (upper bound) that limits the number of clients it can serve and a lower bound on the number of clients it must serve if it is opened. We develop an LP rounding framework that exploits a Voronoi diagram-based clustering approach to derive the first bicriteria constant approximation algorithm for this problem with non-uniform lower bounds and uniform upper bounds. This naturally leads to the the first LP-based approximation algorithm for the lower bounded facility location problem (with non-uniform lower bounds).
We also demonstrate the versatility of our framework by extending this and presenting the first constant approximation algorithm for some connected variant of the problems in which the facilities are required to be connected as well.

Zachary Friggstad, Mohsen Rezapour, and Mohammad R. Salavatipour. Approximating Connected Facility Location with Lower and Upper Bounds via LP Rounding. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 1:1-1:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{friggstad_et_al:LIPIcs.SWAT.2016.1, author = {Friggstad, Zachary and Rezapour, Mohsen and Salavatipour, Mohammad R.}, title = {{Approximating Connected Facility Location with Lower and Upper Bounds via LP Rounding}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {1:1--1:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.1}, URN = {urn:nbn:de:0030-drops-60302}, doi = {10.4230/LIPIcs.SWAT.2016.1}, annote = {Keywords: Facility Location, Approximation Algorithm, LP Rounding} }

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