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

We consider the Max--Section problem, where we are given an undirected graph G=(V,E)equipped with non-negative edge weights w: E → R_+ and the goal is to find a partition of V into three equisized parts while maximizing the total weight of edges crossing between different parts. Max-3-Section is closely related to other well-studied graph partitioning problems, e.g., Max-Cut, Max-3-Cut, and Max-Bisection. We present a polynomial time algorithm achieving an approximation of 0.795, that improves upon the previous best known approximation of 0.673. The requirement of multiple parts that have equal sizes renders Max-3-Section much harder to cope with compared to, e.g., Max-Bisection. We show a new algorithm that combines the existing approach of Lassere hierarchy along with a random cut strategy that suffices to give our result.

Dor Katzelnick, Aditya Pillai, Roy Schwartz, and Mohit Singh. An Improved Approximation Algorithm for the Max-3-Section Problem. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 69:1-69:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{katzelnick_et_al:LIPIcs.ESA.2023.69, author = {Katzelnick, Dor and Pillai, Aditya and Schwartz, Roy and Singh, Mohit}, title = {{An Improved Approximation Algorithm for the Max-3-Section Problem}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {69:1--69:17}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.69}, URN = {urn:nbn:de:0030-drops-187229}, doi = {10.4230/LIPIcs.ESA.2023.69}, annote = {Keywords: Approximation Algorithms, Semidefinite Programming, Max-Cut, Max-Bisection} }

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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

Motivated by the use of high speed circuit switches in large scale data centers, we consider the problem of circuit switch scheduling. In this problem we are given demands between pairs of servers and the goal is to schedule at every time step a matching between the servers while maximizing the total satisfied demand over time. The crux of this scheduling problem is that once one shifts from one matching to a different one a fixed delay delta is incurred during which no data can be transmitted.
For the offline version of the problem we present a (1-(1/e)-epsilon) approximation ratio (for any constant epsilon >0). Since the natural linear programming relaxation for the problem has an unbounded integrality gap, we adopt a hybrid approach that combines the combinatorial greedy with randomized rounding of a different suitable linear program. For the online version of the problem we present a (bi-criteria) ((e-1)/(2e-1)-epsilon)-competitive ratio (for any constant epsilon >0 ) that exceeds time by an additive factor of O(delta/epsilon). We note that no uni-criteria online algorithm is possible. Surprisingly, we obtain the result by reducing the online version to the offline one.

Roy Schwartz, Mohit Singh, and Sina Yazdanbod. Online and Offline Algorithms for Circuit Switch Scheduling. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{schwartz_et_al:LIPIcs.FSTTCS.2019.27, author = {Schwartz, Roy and Singh, Mohit and Yazdanbod, Sina}, title = {{Online and Offline Algorithms for Circuit Switch Scheduling}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {27:1--27:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.27}, URN = {urn:nbn:de:0030-drops-115893}, doi = {10.4230/LIPIcs.FSTTCS.2019.27}, annote = {Keywords: approximation algorithm, online, matching, scheduling} }

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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

We study the problem of allocating m items to n agents subject to maximizing the Nash social welfare (NSW) objective. We write a novel convex programming relaxation for this problem, and we show that a simple randomized rounding algorithm gives a 1/e approximation factor of the objective, breaking the 1/2e^(1/e) approximation factor of Cole and Gkatzelis.
Our main technical contribution is an extension of Gurvits's lower bound on the coefficient of the square-free monomial of a degree m-homogeneous stable polynomial on m variables to all homogeneous polynomials. We use this extension to analyze the expected welfare of the allocation returned by our randomized rounding algorithm.

Nima Anari, Shayan Oveis Gharan, Amin Saberi, and Mohit Singh. Nash Social Welfare, Matrix Permanent, and Stable Polynomials. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 36:1-36:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{anari_et_al:LIPIcs.ITCS.2017.36, author = {Anari, Nima and Oveis Gharan, Shayan and Saberi, Amin and Singh, Mohit}, title = {{Nash Social Welfare, Matrix Permanent, and Stable Polynomials}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {36:1--36:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.36}, URN = {urn:nbn:de:0030-drops-81489}, doi = {10.4230/LIPIcs.ITCS.2017.36}, annote = {Keywords: Nash Welfare, Permanent, Matching, Stable Polynomial, Randomized Algorithm, Saddle Point} }

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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

We present a Gaussian random walk in a polytope that starts at a point inside and continues until it gets absorbed at a vertex. Our main result is that the probability distribution induced on the
vertices by this random walk has strong negative dependence properties for matroid polytopes. Such distributions are highly sought after in randomized algorithms as they imply concentration
properties. Our random walk is simple to implement, computationally efficient and can be viewed as an algorithm to round the starting point in an unbiased manner. The proof relies on a simple
inductive argument that synthesizes the combinatorial structure of matroid polytopes with the geometric structure of multivariate Gaussian distributions. Our result not only implies a long
line of past results in a unified and transparent manner, but also implies new results about constructing negatively associated distributions for all matroids.

Yuval Peres, Mohit Singh, and Nisheeth K. Vishnoi. Random Walks in Polytopes and Negative Dependence. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 50:1-50:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{peres_et_al:LIPIcs.ITCS.2017.50, author = {Peres, Yuval and Singh, Mohit and Vishnoi, Nisheeth K.}, title = {{Random Walks in Polytopes and Negative Dependence}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {50:1--50:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.50}, URN = {urn:nbn:de:0030-drops-81464}, doi = {10.4230/LIPIcs.ITCS.2017.50}, annote = {Keywords: Random walks, Matroid, Polytope, Brownian motion, Negative dependence} }

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**Published in:** LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

We study weighted bipartite edge coloring problem, which is a generalization of two classical problems: bin packing and edge coloring. This problem has been inspired from the study of Clos networks in multirate switching environment in communication networks. In weighted bipartite edge coloring problem, we are given an edge-weighted bipartite multi-graph G=(V,E) with weights
w:E\rightarrow [0,1]. The goal is to find a proper
weighted coloring of the edges with as few colors as possible. An
edge coloring of the weighted graph is called a proper
weighted coloring if the sum of the weights of the edges incident
to a vertex of any color is at most one. Chung and Ross conjectured 2m-1 colors are sufficient for a proper weighted coloring, where m denotes the minimum number of unit sized bins needed to pack the weights of all edges incident at any vertex. We give an algorithm that returns a coloring with at most \lceil 2.2223m \rceil colors improving on the previous result of \frac{9m}{4} by Feige and Singh.
Our algorithm is purely combinatorial and combines the König's theorem for edge coloring bipartite graphs and first-fit decreasing heuristic for bin packing. However, our analysis uses configuration linear program for the bin packing problem to give the improved result.

Arindam Khan and Mohit Singh. On Weighted Bipartite Edge Coloring. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 136-150, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{khan_et_al:LIPIcs.FSTTCS.2015.136, author = {Khan, Arindam and Singh, Mohit}, title = {{On Weighted Bipartite Edge Coloring}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {136--150}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.136}, URN = {urn:nbn:de:0030-drops-56437}, doi = {10.4230/LIPIcs.FSTTCS.2015.136}, annote = {Keywords: Edge coloring, Bin packing, Clos networks, Approximation algorithms, Graph algorithms} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Spencer's theorem asserts that, for any family of n subsets of ground set of size n, the elements of the ground set can be "colored" by the values +1 or -1 such that the sum of every set is O(sqrt(n)) in absolute value. All existing proofs of this result recursively construct "partial colorings", which assign +1 or -1 values to half of the ground set. We devise the first algorithm for Spencer's theorem that directly computes a coloring, without recursively computing partial colorings.

Nicholas J. A. Harvey, Roy Schwartz, and Mohit Singh. Discrepancy Without Partial Colorings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 258-273, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{harvey_et_al:LIPIcs.APPROX-RANDOM.2014.258, author = {Harvey, Nicholas J. A. and Schwartz, Roy and Singh, Mohit}, title = {{Discrepancy Without Partial Colorings}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {258--273}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.258}, URN = {urn:nbn:de:0030-drops-47014}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.258}, annote = {Keywords: Combinatorial Discrepancy, Brownian Motion, Semi-Definite Programming, Randomized Algorithm} }

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