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**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

We present approximation algorithms for the Fault-tolerant k-Supplier with Outliers (FkSO) problem. This is a common generalization of two known problems - k-Supplier with Outliers, and Fault-tolerant k-Supplier - each of which generalize the well-known k-Supplier problem. In the k-Supplier problem the goal is to serve n clients C, by opening k facilities from a set of possible facilities F; the objective function is the farthest that any client must travel to access an open facility. In FkSO, each client v has a fault-tolerance 𝓁_v, and now desires 𝓁_v facilities to serve it; so each client v’s contribution to the objective function is now its distance to the 𝓁_v^th closest open facility. Furthermore, we are allowed to choose m clients that we will serve, and only those clients contribute to the objective function, while the remaining n-m are considered outliers.
Our main result is a (4t-1)-approximation for the FkSO problem, where t is the number of distinct values of 𝓁_v that appear in the instance. At t = 1, i.e. in the case where the 𝓁_v’s are uniformly some 𝓁, this yields a 3-approximation, improving upon the 11-approximation given for the uniform case by Inamdar and Varadarajan [2020], who also introduced the problem. Our result for the uniform case matches tight 3-approximations that exist for k-Supplier, k-Supplier with Outliers, and Fault-tolerant k-Supplier.
Our key technical contribution is an application of the round-or-cut schema to FkSO. Guided by an LP relaxation, we reduce to a simpler optimization problem, which we can solve to obtain distance bounds for the "round" step, and valid inequalities for the "cut" step. By varying how we reduce to the simpler problem, we get varying distance bounds - we include a variant that gives a (2^t + 1)-approximation, which is better for t ∈ {2,3}. In addition, for t = 1, we give a more straightforward application of round-or-cut, yielding a 3-approximation that is much simpler than our general algorithm.

Deeparnab Chakrabarty, Luc Cote, and Ankita Sarkar. Fault-tolerant k-Supplier with Outliers. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 23:1-23:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chakrabarty_et_al:LIPIcs.STACS.2024.23, author = {Chakrabarty, Deeparnab and Cote, Luc and Sarkar, Ankita}, title = {{Fault-tolerant k-Supplier with Outliers}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {23:1--23:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.23}, URN = {urn:nbn:de:0030-drops-197336}, doi = {10.4230/LIPIcs.STACS.2024.23}, annote = {Keywords: Clustering, approximation algorithms, round-or-cut} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

In this paper, we consider center-based clustering problems where C, the set of points to be clustered, lies in a metric space (X,d), and the set X of candidate centers is potentially infinite-sized. We call such problems continuous clustering problems to differentiate them from the discrete clustering problems where the set of candidate centers is explicitly given. It is known that for many objectives, when one restricts the set of centers to C itself and applies an α_dis-approximation algorithm for the discrete version, one obtains a β ⋅ α_{dis}-approximation algorithm for the continuous version via the triangle inequality property of the distance function. Here β depends on the objective, and for many objectives such as k-median, β = 2, while for some others such as k-means, β = 4. The motivating question in this paper is whether this gap of factor β between continuous and discrete problems is inherent, or can one design better algorithms for continuous clustering than simply reducing to the discrete case as mentioned above? In a recent SODA 2021 paper, Cohen-Addad, Karthik, and Lee prove a factor-2 and a factor-4 hardness, respectively, for the continuous versions of the k-median and k-means problems, even when the number of cluster centers is a constant. The discrete problem for a constant number of centers is easily solvable exactly using enumeration, and therefore, in certain regimes, the "β-factor loss" seems unavoidable.
In this paper, we describe a technique based on the round-or-cut framework to approach continuous clustering problems. We show that, for the continuous versions of some clustering problems, we can design approximation algorithms attaining a better factor than the β-factor blow-up mentioned above. In particular, we do so for: the uncapacitated facility location problem with uniform facility opening costs (λ-UFL); the k-means problem; the individually fair k-median problem; and the k-center with outliers problem. Notably, for λ-UFL, where β = 2 and the discrete version is NP-hard to approximate within a factor of 1.27, we describe a 2.32-approximation for the continuous version, and indeed 2.32 < 2 × 1.27. Also, for k-means, where β = 4 and the best known approximation factor for the discrete version is 9, we obtain a 32-approximation for the continuous version, which is better than 4 × 9 = 36.
The main challenge one faces is that most algorithms for the discrete clustering problems, including the state of the art solutions, depend on Linear Program (LP) relaxations that become infinite-sized in the continuous version. To overcome this, we design new linear program relaxations for the continuous clustering problems which, although having exponentially many constraints, are amenable to the round-or-cut framework.

Deeparnab Chakrabarty, Maryam Negahbani, and Ankita Sarkar. Approximation Algorithms for Continuous Clustering and Facility Location Problems. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chakrabarty_et_al:LIPIcs.ESA.2022.33, author = {Chakrabarty, Deeparnab and Negahbani, Maryam and Sarkar, Ankita}, title = {{Approximation Algorithms for Continuous Clustering and Facility Location Problems}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {33:1--33:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.33}, URN = {urn:nbn:de:0030-drops-169710}, doi = {10.4230/LIPIcs.ESA.2022.33}, annote = {Keywords: Approximation Algorithms, Clustering, Facility Location, Fairness, Outliers} }

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**Published in:** LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

We consider the hospital-residents problem where both hospitals and residents can have lower quotas. The input is a bipartite graph G = (ℛ∪ℋ,E), each vertex in ℛ∪ℋ has a strict preference ordering over its neighbors. The sets ℛ and ℋ denote the sets of residents and hospitals respectively. Each hospital has an upper and a lower quota denoting the maximum and minimum number of residents that can be assigned to it. Residents have upper quota equal to one, however, there may be a requirement that some residents must not be left unassigned in the output matching. We call this as the residents' lower quota.
We show that whenever the set of matchings satisfying all the lower and upper quotas is non-empty, there always exists a matching that is popular among the matchings in this set. We give a polynomial-time algorithm to compute such a matching.

Meghana Nasre, Prajakta Nimbhorkar, Keshav Ranjan, and Ankita Sarkar. Popular Matchings in the Hospital-Residents Problem with Two-Sided Lower Quotas. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 30:1-30:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{nasre_et_al:LIPIcs.FSTTCS.2021.30, author = {Nasre, Meghana and Nimbhorkar, Prajakta and Ranjan, Keshav and Sarkar, Ankita}, title = {{Popular Matchings in the Hospital-Residents Problem with Two-Sided Lower Quotas}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {30:1--30:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-215-0}, ISSN = {1868-8969}, year = {2021}, volume = {213}, editor = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.30}, URN = {urn:nbn:de:0030-drops-155419}, doi = {10.4230/LIPIcs.FSTTCS.2021.30}, annote = {Keywords: Matching, Popularity, Lower quota, Preferences} }