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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

In this paper we study two fully-dynamic multi-dimensional vector load balancing problems with recourse. The adversary presents a stream of n job insertions and deletions, where each job j is a vector in ℝ^d_{≥ 0}. In the vector scheduling problem, the algorithm must maintain an assignment of the active jobs to m identical machines to minimize the makespan (maximum load on any dimension on any machine). In the vector bin packing problem, the algorithm must maintain an assignment of active jobs into a number of bins of unit capacity in all dimensions, to minimize the number of bins currently used. In both problems, the goal is to maintain solutions that are competitive against the optimal solution for the active set of jobs, at every time instant. The algorithm is allowed to change the assignment from time to time, with the secondary objective of minimizing the amortized recourse, which is the average cardinality of the change of the assignment per update to the instance.
For the vector scheduling problem, we present two simple algorithms. The first is a randomized algorithm with an O(1) amortized recourse and an O(log d/log log d) competitive ratio against oblivious adversaries. The second algorithm is a deterministic algorithm that is competitive against adaptive adversaries but with a slightly higher competitive ratio of O(log d) and a per-job recourse guarantee bounded by Õ(log n + log d log OPT). We also prove a sharper instance-dependent recourse guarantee for the deterministic algorithm.
For the vector bin packing problem, we make the so-called small jobs assumption that the size of all jobs in all the coordinates is O(1/log d) and present a simple O(1)-competitive algorithm with O(log n) recourse against oblivious adversaries.
For both problems, the main challenge is to determine when and how to migrate jobs to maintain competitive solutions. Our central idea is that for each job, we make these decisions based only on the active set of jobs that are "earlier" than this job in some ordering ≺ of the jobs.

Varun Gupta, Ravishankar Krishnaswamy, Sai Sandeep, and Janani Sundaresan. Look Before, Before You Leap: Online Vector Load Balancing with Few Reassignments. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 65:1-65:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gupta_et_al:LIPIcs.ITCS.2023.65, author = {Gupta, Varun and Krishnaswamy, Ravishankar and Sandeep, Sai and Sundaresan, Janani}, title = {{Look Before, Before You Leap: Online Vector Load Balancing with Few Reassignments}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {65:1--65:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.65}, URN = {urn:nbn:de:0030-drops-175685}, doi = {10.4230/LIPIcs.ITCS.2023.65}, annote = {Keywords: Vector Scheduling, Vector Load Balancing} }

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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

We consider the online carpooling problem: given n vertices, a sequence of edges arrive over time. When an edge e_t = (u_t, v_t) arrives at time step t, the algorithm must orient the edge either as v_t → u_t or u_t → v_t, with the objective of minimizing the maximum discrepancy of any vertex, i.e., the absolute difference between its in-degree and out-degree. Edges correspond to pairs of persons wanting to ride together, and orienting denotes designating the driver. The discrepancy objective then corresponds to every person driving close to their fair share of rides they participate in.
In this paper, we design efficient algorithms which can maintain polylog(n,T) maximum discrepancy (w.h.p) over any sequence of T arrivals, when the arriving edges are sampled independently and uniformly from any given graph G. This provides the first polylogarithmic bounds for the online (stochastic) carpooling problem. Prior to this work, the best known bounds were O(√{n log n})-discrepancy for any adversarial sequence of arrivals, or O(log log n)-discrepancy bounds for the stochastic arrivals when G is the complete graph.
The technical crux of our paper is in showing that the simple greedy algorithm, which has provably good discrepancy bounds when the arriving edges are drawn uniformly at random from the complete graph, also has polylog discrepancy when G is an expander graph. We then combine this with known expander-decomposition results to design our overall algorithm.

Anupam Gupta, Ravishankar Krishnaswamy, Amit Kumar, and Sahil Singla. Online Carpooling Using Expander Decompositions. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gupta_et_al:LIPIcs.FSTTCS.2020.23, author = {Gupta, Anupam and Krishnaswamy, Ravishankar and Kumar, Amit and Singla, Sahil}, title = {{Online Carpooling Using Expander Decompositions}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {23:1--23:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.23}, URN = {urn:nbn:de:0030-drops-132647}, doi = {10.4230/LIPIcs.FSTTCS.2020.23}, annote = {Keywords: Online Algorithms, Discrepancy Minimization, Carpooling} }

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APPROX

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

In this paper, we study the online metric matching with recourse (OMM-Recourse) problem. Given a metric space with k servers, a sequence of clients is revealed online. A client must be matched to an available server on arrival. Unlike the classical online matching model where the match is irrevocable, the recourse model permits the algorithm to rematch existing clients upon the arrival of a new client. The goal is to maintain an online matching with a near-optimal total cost, while at the same time not rematching too many clients.
For the classical online metric matching problem without recourse, the optimal competitive ratio for deterministic algorithms is 2k-1, and the best-known randomized algorithms have competitive ratio O(log² k). For the much-studied special case of line metric, the best-known algorithms have competitive ratios of O(log k). Improving these competitive ratios (or showing lower bounds) are important open problems in this line of work.
In this paper, we show that logarithmic recourse significantly improves the quality of matchings we can maintain online. For general metrics, we show a deterministic O(log k)-competitive algorithm, with O(log k) recourse per client, an exponential improvement over the 2k-1 lower bound without recourse. For line metrics we show a deterministic 3-competitive algorithm with O(log k) amortized recourse, again improving the best-known O(log k)-competitive algorithms without recourse. The first result (general metrics) simulates a batched version of the classical algorithm for OMM called Permutation. The second result (line metric) also uses Permutation as the foundation but makes non-trivial changes to the matching to balance the competitive ratio and recourse.
Finally, we also consider the model when both clients and servers may arrive or depart dynamically, and exhibit a simple randomized O(log n)-competitive algorithm with O(log Δ) recourse, where n and Δ are the number of points and the aspect ratio of the underlying metric. We remark that no non-trivial bounds are possible in this fully-dynamic model when no recourse is allowed.

Varun Gupta, Ravishankar Krishnaswamy, and Sai Sandeep. Permutation Strikes Back: The Power of Recourse in Online Metric Matching. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 40:1-40:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gupta_et_al:LIPIcs.APPROX/RANDOM.2020.40, author = {Gupta, Varun and Krishnaswamy, Ravishankar and Sandeep, Sai}, title = {{Permutation Strikes Back: The Power of Recourse in Online Metric Matching}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {40:1--40:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.40}, URN = {urn:nbn:de:0030-drops-126431}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.40}, annote = {Keywords: online algorithms, bipartite matching} }

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APPROX

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

In this paper, we introduce and study the Robust-Correlation-Clustering problem: given a graph G = (V,E) where every edge is either labeled + or - (denoting similar or dissimilar pairs of vertices), and a parameter m, the goal is to delete a set D of m vertices, and partition the remaining vertices V \ D into clusters to minimize the cost of the clustering, which is the sum of the number of + edges with end-points in different clusters and the number of - edges with end-points in the same cluster. This generalizes the classical Correlation-Clustering problem which is the special case when m = 0. Correlation clustering is useful when we have (only) qualitative information about the similarity or dissimilarity of pairs of points, and Robust-Correlation-Clustering equips this model with the capability to handle noise in datasets.
In this work, we present a constant-factor bi-criteria algorithm for Robust-Correlation-Clustering on complete graphs (where our solution is O(1)-approximate w.r.t the cost while however discarding O(1) m points as outliers), and also complement this by showing that no finite approximation is possible if we do not violate the outlier budget. Our algorithm is very simple in that it first does a simple LP-based pre-processing to delete O(m) vertices, and subsequently runs a particular Correlation-Clustering algorithm ACNAlg [Ailon et al., 2005] on the residual instance. We then consider general graphs, and show (O(log n), O(log^2 n)) bi-criteria algorithms while also showing a hardness of alpha_MC on both the cost and the outlier violation, where alpha_MC is the lower bound for the Minimum-Multicut problem.

Devvrit, Ravishankar Krishnaswamy, and Nived Rajaraman. Robust Correlation Clustering. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 33:1-33:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{devvrit_et_al:LIPIcs.APPROX-RANDOM.2019.33, author = {Devvrit and Krishnaswamy, Ravishankar and Rajaraman, Nived}, title = {{Robust Correlation Clustering}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {33:1--33:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.33}, URN = {urn:nbn:de:0030-drops-112483}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.33}, annote = {Keywords: Correlation Clustering, Outlier Detection, Clustering, Approximation Algorithms} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

In this paper, we introduce and study the Non-Uniform k-Center (NUkC) problem. Given a finite metric space (X, d) and a collection of balls of radii {r_1 >= ... >= r_k}, the NUkC problem is to find a placement of their centers on the metric space and find the minimum dilation alpha, such that the union of balls of radius alpha*r_i around the i-th center covers all the points in X. This problem naturally arises as a min-max vehicle routing problem with fleets of different speeds, or as a wireless router placement problem with routers of different powers/ranges.
The NUkC problem generalizes the classic k-center problem when all the k radii are the same (which can be assumed to be 1 after scaling). It also generalizes the k-center with outliers (kCwO for short) problem when there are k balls of radius 1 and l balls of radius 0. There are 2-approximation and 3-approximation algorithms known for these problems respectively; the former is best possible unless P=NP and the latter remains unimproved for 15 years.
We first observe that no O(1)-approximation is to the optimal dilation is possible unless P=NP, implying that the NUkC problem is more non-trivial than the above two problems. Our main algorithmic result is an (O(1), O(1))-bi-criteria approximation result: we give an O(1)-approximation to the optimal dilation, however, we may open Theta(1) centers of each radii. Our techniques also allow us to prove a simple (uni-criteria), optimal 2-approximation to the kCwO problem improving upon the long-standing 3-factor. Our main technical contribution is a connection between the NUkC problem and the so-called firefighter problems on trees which have been studied recently in the TCS community. We show NUkC is as hard as the firefighter problem.
While we don't know if the converse is true, we are able to adapt ideas from recent works [Chalermsook/Chuzhoy, SODA 2010; Asjiashvili/Baggio/Zenklusen, arXiv 2016] in non-trivial ways to obtain our constant factor bi-criteria approximation.

Deeparnab Chakrabarty, Prachi Goyal, and Ravishankar Krishnaswamy. The Non-Uniform k-Center Problem. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 67:1-67:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chakrabarty_et_al:LIPIcs.ICALP.2016.67, author = {Chakrabarty, Deeparnab and Goyal, Prachi and Krishnaswamy, Ravishankar}, title = {{The Non-Uniform k-Center Problem}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {67:1--67:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.67}, URN = {urn:nbn:de:0030-drops-62178}, doi = {10.4230/LIPIcs.ICALP.2016.67}, annote = {Keywords: Clustering, k-Center, Approximation Algorithms, Firefighter Problem} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

We consider a natural online optimization problem set on the real line. The state of the online algorithm at each integer time is a location on the real line. At each integer time, a convex function arrives online. In response, the online algorithm picks a new location. The cost paid by the online algorithm for this response is the distance moved plus the value of the function at the final destination. The objective is then to minimize the aggregate cost over all time. The motivating application is rightsizing power-proportional data centers. We give a 2-competitive algorithm for this problem. We also give a 3-competitive memoryless algorithm, and show that this is the best competitive ratio achievable by a deterministic memoryless algorithm. Finally we show that this online problem is strictly harder than the standard ski rental problem.

Nikhil Bansal, Anupam Gupta, Ravishankar Krishnaswamy, Kirk Pruhs, Kevin Schewior, and Cliff Stein. A 2-Competitive Algorithm For Online Convex Optimization With Switching Costs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 96-109, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bansal_et_al:LIPIcs.APPROX-RANDOM.2015.96, author = {Bansal, Nikhil and Gupta, Anupam and Krishnaswamy, Ravishankar and Pruhs, Kirk and Schewior, Kevin and Stein, Cliff}, title = {{A 2-Competitive Algorithm For Online Convex Optimization With Switching Costs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {96--109}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.96}, URN = {urn:nbn:de:0030-drops-52970}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.96}, annote = {Keywords: Stochastic, Scheduling} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

The Euclidean k-means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of n points in Euclidean space R^d, and the goal is to choose k center points in R^d so that the sum of squared distances of each point to its nearest center is minimized. The best approximation algorithms for this problem include a polynomial time constant factor approximation for general k and a (1+c)-approximation which runs in time poly(n) exp(k/c). At the other extreme, the only known computational complexity result for this problem is NP-hardness [Aloise et al.'09]. The main difficulty in obtaining hardness results stems from the Euclidean nature of the problem, and the fact that any point in R^d can be a potential center. This gap in understanding left open the intriguing possibility that the problem might admit a PTAS for all k, d.
In this paper we provide the first hardness of approximation for the Euclidean k-means problem. Concretely, we show that there exists a constant c > 0 such that it is NP-hard to approximate the k-means objective to within a factor of (1+c). We show this via an efficient reduction from the vertex cover problem on triangle-free graphs: given a triangle-free graph, the goal is to choose the fewest number of vertices which are incident on all the edges. Additionally, we give a proof that the current best hardness results for vertex cover can be carried over to triangle-free graphs. To show this we transform G, a known hard vertex cover instance, by taking a graph product with a suitably chosen graph H, and showing that the size of the (normalized) maximum independent set is almost exactly preserved in the product graph using a spectral analysis, which might be of independent interest.

Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy, and Ali Kemal Sinop. The Hardness of Approximation of Euclidean k-Means. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 754-767, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{awasthi_et_al:LIPIcs.SOCG.2015.754, author = {Awasthi, Pranjal and Charikar, Moses and Krishnaswamy, Ravishankar and Sinop, Ali Kemal}, title = {{The Hardness of Approximation of Euclidean k-Means}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {754--767}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.754}, URN = {urn:nbn:de:0030-drops-51178}, doi = {10.4230/LIPIcs.SOCG.2015.754}, annote = {Keywords: Euclidean k-means, Hardness of Approximation, Vertex Cover} }

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