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**Published in:** LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

For two point sets A, B ⊂ ℝ^d, with |A| = |B| = n and d > 1 a constant, and for a parameter ε > 0, we present a randomized algorithm that, with probability at least 1/2, computes in O(n(ε^{-O(d³)}log log n + ε^{-O(d)}log⁴ nlog⁵log n)) time, an ε-approximate minimum-cost perfect matching under any L_p-metric. All previous algorithms take n(ε^{-1}log n)^{Ω(d)} time. We use a randomly-shifted tree, with a polynomial branching factor and O(log log n) height, to define a tree-based distance function that ε-approximates the L_p metric as well as to compute the matching hierarchically. Then, we apply the primal-dual framework on a compressed representation of the residual graph to obtain an efficient implementation of the Hungarian-search and augment operations.

Pankaj K. Agarwal, Sharath Raghvendra, Pouyan Shirzadian, and Rachita Sowle. An Improved ε-Approximation Algorithm for Geometric Bipartite Matching. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 6:1-6:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{agarwal_et_al:LIPIcs.SWAT.2022.6, author = {Agarwal, Pankaj K. and Raghvendra, Sharath and Shirzadian, Pouyan and Sowle, Rachita}, title = {{An Improved \epsilon-Approximation Algorithm for Geometric Bipartite Matching}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {6:1--6:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.6}, URN = {urn:nbn:de:0030-drops-161660}, doi = {10.4230/LIPIcs.SWAT.2022.6}, annote = {Keywords: Euclidean bipartite matching, approximation algorithms, primal dual method} }

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**Published in:** LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

We provide a novel implementation of the classical Work Function Algorithm (WFA) for the k-server problem. In our implementation, processing a request takes O(n²+k²) time per request; where n is the total number of requests and k is the total number of servers. All prior implementations take Ω(kn² +k³) time per request. Previous approaches process a request by solving a min-cost flow problem. Instead, we show that processing a request can be reduced to an execution of the Dijkstra’s shortest-path algorithm on a carefully computed weighted graph leading to the speed-up.

Sharath Raghvendra and Rachita Sowle. A Scalable Work Function Algorithm for the k-Server Problem. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 30:1-30:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{raghvendra_et_al:LIPIcs.SWAT.2022.30, author = {Raghvendra, Sharath and Sowle, Rachita}, title = {{A Scalable Work Function Algorithm for the k-Server Problem}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {30:1--30:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.30}, URN = {urn:nbn:de:0030-drops-161906}, doi = {10.4230/LIPIcs.SWAT.2022.30}, annote = {Keywords: k-server, Work Function Algorithm, Minimum-cost Flow} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We present a weighted approach to compute a maximum cardinality matching in an arbitrary bipartite graph. Our main result is a new algorithm that takes as input a weighted bipartite graph G(A cup B,E) with edge weights of 0 or 1. Let w <= n be an upper bound on the weight of any matching in G. Consider the subgraph induced by all the edges of G with a weight 0. Suppose every connected component in this subgraph has O(r) vertices and O(mr/n) edges. We present an algorithm to compute a maximum cardinality matching in G in O~(m(sqrt{w} + sqrt{r} + wr/n)) time.
When all the edge weights are 1 (symmetrically when all weights are 0), our algorithm will be identical to the well-known Hopcroft-Karp (HK) algorithm, which runs in O(m sqrt{n}) time. However, if we can carefully assign weights of 0 and 1 on its edges such that both w and r are sub-linear in n and wr=O(n^{gamma}) for gamma < 3/2, then we can compute maximum cardinality matching in G in o(m sqrt{n}) time. Using our algorithm, we obtain a new O~(n^{4/3}/epsilon^4) time algorithm to compute an epsilon-approximate bottleneck matching of A,B subsetR^2 and an 1/(epsilon^{O(d)}}n^{1+(d-1)/(2d-1)}) poly log n time algorithm for computing epsilon-approximate bottleneck matching in d-dimensions. All previous algorithms take Omega(n^{3/2}) time. Given any graph G(A cup B,E) that has an easily computable balanced vertex separator for every subgraph G'(V',E') of size |V'|^{delta}, for delta in [1/2,1), we can apply our algorithm to compute a maximum matching in O~(mn^{delta/1+delta}) time improving upon the O(m sqrt{n}) time taken by the HK-Algorithm.

Nathaniel Lahn and Sharath Raghvendra. A Weighted Approach to the Maximum Cardinality Bipartite Matching Problem with Applications in Geometric Settings. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 48:1-48:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{lahn_et_al:LIPIcs.SoCG.2019.48, author = {Lahn, Nathaniel and Raghvendra, Sharath}, title = {{A Weighted Approach to the Maximum Cardinality Bipartite Matching Problem with Applications in Geometric Settings}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {48:1--48:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.48}, URN = {urn:nbn:de:0030-drops-104528}, doi = {10.4230/LIPIcs.SoCG.2019.48}, annote = {Keywords: Bipartite matching, Bottleneck matching} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

In the online metric bipartite matching problem, we are given a set S of server locations in a metric space. Requests arrive one at a time, and on its arrival, we need to immediately and irrevocably match it to a server at a cost which is equal to the distance between these locations. A alpha-competitive algorithm will assign requests to servers so that the total cost is at most alpha times the cost of M_{Opt} where M_{Opt} is the minimum cost matching between S and R.
We consider this problem in the adversarial model for the case where S and R are points on a line and |S|=|R|=n. We improve the analysis of the deterministic Robust Matching Algorithm (RM-Algorithm, Nayyar and Raghvendra FOCS'17) from O(log^2 n) to an optimal Theta(log n). Previously, only a randomized algorithm under a weaker oblivious adversary achieved a competitive ratio of O(log n) (Gupta and Lewi, ICALP'12). The well-known Work Function Algorithm (WFA) has a competitive ratio of O(n) and Omega(log n) for this problem. Therefore, WFA cannot achieve an asymptotically better competitive ratio than the RM-Algorithm.

Sharath Raghvendra. Optimal Analysis of an Online Algorithm for the Bipartite Matching Problem on a Line. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 67:1-67:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{raghvendra:LIPIcs.SoCG.2018.67, author = {Raghvendra, Sharath}, title = {{Optimal Analysis of an Online Algorithm for the Bipartite Matching Problem on a Line}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {67:1--67:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.67}, URN = {urn:nbn:de:0030-drops-87803}, doi = {10.4230/LIPIcs.SoCG.2018.67}, annote = {Keywords: Bipartite Matching, Online Algorithms, Adversarial Model, Line Metric} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Rips complexes are important structures for analyzing topological features of metric spaces. Unfortunately, generating these complexes constitutes an expensive task because of a combinatorial explosion in the complex size. For n points in R^d, we present a scheme to construct a 4.24-approximation of the multi-scale filtration of the Rips complex in the L-infinity metric, which extends to a O(d^{0.25})-approximation of the Rips filtration for the Euclidean case. The k-skeleton of the resulting approximation has a total size of n2^{O(d log k)}. The scheme is based on the integer lattice and on the barycentric subdivision of the d-cube.

Aruni Choudhary, Michael Kerber, and Sharath Raghvendra. Improved Approximate Rips Filtrations with Shifted Integer Lattices. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{choudhary_et_al:LIPIcs.ESA.2017.28, author = {Choudhary, Aruni and Kerber, Michael and Raghvendra, Sharath}, title = {{Improved Approximate Rips Filtrations with Shifted Integer Lattices}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {28:1--28:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.28}, URN = {urn:nbn:de:0030-drops-78259}, doi = {10.4230/LIPIcs.ESA.2017.28}, annote = {Keywords: Persistent homology, Rips filtrations, Approximation algorithms, Topological Data Analysis} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

We study the Online Minimum Metric Bipartite Matching Problem. In this problem, we are given point sets S and R which correspond to the server and request locations; here |S|=|R|=n. All these locations are points from some metric space and the cost of matching a server to a request is given by the distance between their locations in this space. In this problem, the request points arrive one at a time. When a request arrives, we must immediately and irrevocably match it to a "free" server. The matching obtained after all the requests are processed is the online matching M. The cost of M is the sum of the cost of its edges. The performance of any online algorithm is the worst-case ratio of the cost of its online solution M to the minimum-cost matching.
We present a deterministic online algorithm for this problem. Our algorithm is the first to simultaneously achieve optimal performances in the well-known adversarial and the random arrival models. For the adversarial model, we obtain a competitive ratio of 2n-1 + o(1); it is known that no deterministic algorithm can do better than 2n-1. In the random arrival model, our algorithm obtains a competitive ratio of 2H_n - 1 + o(1); where H_n is the n-th Harmonic number. We also prove that any online algorithm will have a competitive ratio of at least 2H_n - 1-o(1) in this model.
We use a new variation of the offline primal-dual method for computing minimum cost matching to compute the online matching. Our primal-dual method is based on a relaxed linear-program. Under metric costs, this specific relaxation helps us relate the cost of the online matching with the offline matching leading to its robust properties.

Sharath Raghvendra. A Robust and Optimal Online Algorithm for Minimum Metric Bipartite Matching. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{raghvendra:LIPIcs.APPROX-RANDOM.2016.18, author = {Raghvendra, Sharath}, title = {{A Robust and Optimal Online Algorithm for Minimum Metric Bipartite Matching}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {18:1--18:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.18}, URN = {urn:nbn:de:0030-drops-66410}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.18}, annote = {Keywords: Online Algorithms, Metric Bipartite Matching} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Classical methods to model topological properties of point clouds, such as the Vietoris-Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips complex with improved bounds for size: precisely, for n points in R^d, we obtain a O(d)-approximation with at most n2^{O(d log k)} simplices of dimension k or lower. In conjunction with dimension reduction techniques, our approach yields a O(polylog (n))-approximation of size n^{O(1)} for Rips filtrations on arbitrary metric spaces. This result stems from high-dimensional lattice geometry and exploits properties of the permutahedral lattice, a well-studied structure in discrete geometry.
Building on the same geometric concept, we also present a lower bound result on the size of an approximate filtration: we construct a point set for which every (1+epsilon)-approximation of the Cech filtration has to contain n^{Omega(log log n)} features, provided that epsilon < 1/(log^{1+c}n) for c in (0,1).

Aruni Choudhary, Michael Kerber, and Sharath Raghvendra. Polynomial-Sized Topological Approximations Using the Permutahedron. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{choudhary_et_al:LIPIcs.SoCG.2016.31, author = {Choudhary, Aruni and Kerber, Michael and Raghvendra, Sharath}, title = {{Polynomial-Sized Topological Approximations Using the Permutahedron}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {31:1--31:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.31}, URN = {urn:nbn:de:0030-drops-59236}, doi = {10.4230/LIPIcs.SoCG.2016.31}, annote = {Keywords: Persistent Homology, Topological Data Analysis, Simplicial Approximation, Permutahedron, Approximation Algorithms} }