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

Graph partitioning problems are a central topic of study in algorithms and complexity theory. Edge expansion and vertex expansion, two popular graph partitioning objectives, seek a 2-partition of the vertex set of the graph that minimizes the considered objective. However, for many natural applications, one might require a graph to be partitioned into k parts, for some k >=slant 2. For a k-partition S_1, ..., S_k of the vertex set of a graph G = (V,E), the k-way edge expansion (resp. vertex expansion) of {S_1, ..., S_k} is defined as max_{i in [k]} Phi(S_i), and the balanced k-way edge expansion (resp. vertex expansion) of G is defined as min_{{S_1, ..., S_k} in P_k} max_{i in [k]} Phi(S_i) , where P_k is the set of all balanced k-partitions of V (i.e each part of a k-partition in P_k should have cardinality |V|/k), and Phi(S) denotes the edge expansion (resp. vertex expansion) of S subset V. We study a natural planted model for graphs where the vertex set of a graph has a k-partition S_1, ..., S_k such that the graph induced on each S_i has large expansion, but each S_i has small edge expansion (resp. vertex expansion) in the graph. We give bi-criteria approximation algorithms for computing the balanced k-way edge expansion (resp. vertex expansion) of instances in this planted model.

Anand Louis and Rakesh Venkat. Planted Models for k-Way Edge and Vertex Expansion. 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. 23:1-23:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{louis_et_al:LIPIcs.FSTTCS.2019.23, author = {Louis, Anand and Venkat, Rakesh}, title = {{Planted Models for k-Way Edge and Vertex Expansion}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {23:1--23:15}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.23}, URN = {urn:nbn:de:0030-drops-115857}, doi = {10.4230/LIPIcs.FSTTCS.2019.23}, annote = {Keywords: Vertex Expansion, k-way partitioning, Semi-Random models, Planted Models, Approximation Algorithms, Beyond Worst Case Analysis} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

The problem of computing the vertex expansion of a graph is an NP-hard problem. The current best worst-case approximation guarantees for computing the vertex expansion of a graph are a O(sqrt{log n})-approximation algorithm due to Feige et al. [Uriel Feige et al., 2008], and O(sqrt{OPT log d}) bound in graphs having vertex degrees at most d due to Louis et al. [Louis et al., 2013].
We study a natural semi-random model of graphs with sparse vertex cuts. For certain ranges of parameters, we give an algorithm to recover the planted sparse vertex cut exactly. For a larger range of parameters, we give a constant factor bi-criteria approximation algorithm to compute the graph's balanced vertex expansion. Our algorithms are based on studying a semidefinite programming relaxation for the balanced vertex expansion of the graph.
In addition to being a family of instances that will help us to better understand the complexity of the computation of vertex expansion, our model can also be used in the study of community detection where only a few nodes from each community interact with nodes from other communities. There has been a lot of work on studying random and semi-random graphs with planted sparse edge cuts. To the best of our knowledge, our model of semi-random graphs with planted sparse vertex cuts has not been studied before.

Anand Louis and Rakesh Venkat. Semi-random Graphs with Planted Sparse Vertex Cuts: Algorithms for Exact and Approximate Recovery. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 101:1-101:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{louis_et_al:LIPIcs.ICALP.2018.101, author = {Louis, Anand and Venkat, Rakesh}, title = {{Semi-random Graphs with Planted Sparse Vertex Cuts: Algorithms for Exact and Approximate Recovery}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {101:1--101:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.101}, URN = {urn:nbn:de:0030-drops-91057}, doi = {10.4230/LIPIcs.ICALP.2018.101}, annote = {Keywords: Semi-Random models, Vertex Expansion, Approximation Algorithms, Beyond Worst Case Analysis} }

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

We investigate the value of parallel repetition of one-round games with any number of players k>=2. It has been an open question whether an analogue of Raz's Parallel Repetition Theorem holds for games with more than two players, i.e., whether the value of the repeated game decays exponentially with the number of repetitions. Verbitsky has shown, via a reduction to the density Hales-Jewett theorem, that the value of the repeated game must approach zero, as the number of repetitions increases. However, the rate of decay obtained in this way is extremely slow, and it is an open question whether the true rate is exponential as is the case for all two-player games.
Exponential decay bounds are known for several special cases of multi-player games, e.g., free games and anchored games. In this work, we identify a certain expansion property of the base game and show all games with this property satisfy an exponential decay parallel repetition bound. Free games and anchored games satisfy this expansion property, and thus our parallel repetition theorem reproduces all earlier exponential-decay bounds for multiplayer games. More generally, our parallel repetition bound applies to all multiplayer games that are *connected* in a certain sense.
We also describe a very simple game, called the GHZ game, that does not satisfy this connectivity property, and for which we do not know an exponential decay bound. We suspect that progress on bounding the value of this the parallel repetition of the GHZ game will lead to further progress on the general question.

Irit Dinur, Prahladh Harsha, Rakesh Venkat, and Henry Yuen. Multiplayer Parallel Repetition for Expanding Games. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 37:1-37:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dinur_et_al:LIPIcs.ITCS.2017.37, author = {Dinur, Irit and Harsha, Prahladh and Venkat, Rakesh and Yuen, Henry}, title = {{Multiplayer Parallel Repetition for Expanding Games}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {37:1--37:16}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.37}, URN = {urn:nbn:de:0030-drops-81575}, doi = {10.4230/LIPIcs.ITCS.2017.37}, annote = {Keywords: Parallel Repetition, Multi-player, Expander} }

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

We consider the problem of embedding a finite set of points x_1, ... , x_n in R^d that satisfy l_2^2 triangle inequalities into l_1, when the points are approximately low-dimensional. Goemans (unpublished, appears in a work of Magen and Moharammi (2008) ) showed that such points residing in exactly d dimensions can be embedded into l_1 with distortion at most sqrt{d}. We prove the following robust analogue of this statement: if there exists a r-dimensional subspace Pi such that the projections onto this subspace satisfy sum_{i,j in [n]} norm{Pi x_i - Pi x_j}_2^2 >= Omega(1) * sum_{i,j \in [n]} norm{x_i - x_j}_2^2, then there is an embedding of the points into l_1 with O(sqrt{r}) average distortion. A consequence of this result is that the integrality gap of the well-known Goemans-Linial SDP relaxation for the Uniform Sparsest Cut problem is O(sqrt{r}) on graphs G whose r-th smallest normalized eigenvalue of the Laplacian satisfies lambda_r(G)/n >= Omega(1)*Phi_{SDP}(G). Our result improves upon the previously known bound of O(r) on the average distortion, and the integrality gap of the Goemans-Linial SDP under the same preconditions, proven in [Deshpande and Venkat, 2014], and [Deshpande, Harsha and Venkat 2016].

Yuval Rabani and Rakesh Venkat. Approximating Sparsest Cut in Low Rank Graphs via Embeddings from Approximately Low Dimensional Spaces. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 21:1-21:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{rabani_et_al:LIPIcs.APPROX-RANDOM.2017.21, author = {Rabani, Yuval and Venkat, Rakesh}, title = {{Approximating Sparsest Cut in Low Rank Graphs via Embeddings from Approximately Low Dimensional Spaces}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {21:1--21:14}, 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.2017.21}, URN = {urn:nbn:de:0030-drops-75705}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.21}, annote = {Keywords: Metric Embeddings, Sparsest Cut, Negative type metrics, Approximation Algorithms} }

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**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Goemans showed that any n points x_1,..., x_n in d-dimensions satisfying l_2^2 triangle inequalities can be embedded into l_{1}, with worst-case distortion at most sqrt{d}. We consider an extension of this theorem to the case when the points are approximately low-dimensional as opposed to exactly low-dimensional, and prove the following analogous theorem, albeit with average distortion guarantees: There exists an l_{2}^{2}-to-l_{1} embedding with average distortion at most the stable rank, sr(M), of the matrix M consisting of columns {x_i-x_j}_{i<j}. Average distortion embedding suffices for applications such as the SPARSEST CUT problem. Our embedding gives an approximation algorithm for the SPARSEST CUT problem on low threshold-rank graphs, where earlier work was inspired by Lasserre SDP hierarchy, and improves on a previous result of the first and third author [Deshpande and Venkat, in Proc. 17th APPROX, 2014]. Our ideas give a new perspective on l_{2}^{2} metric, an alternate proof of Goemans' theorem, and a simpler proof for average distortion sqrt{d}.

Amit Deshpande, Prahladh Harsha, and Rakesh Venkat. Embedding Approximately Low-Dimensional l_2^2 Metrics into l_1. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 10:1-10:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{deshpande_et_al:LIPIcs.FSTTCS.2016.10, author = {Deshpande, Amit and Harsha, Prahladh and Venkat, Rakesh}, title = {{Embedding Approximately Low-Dimensional l\underline2^2 Metrics into l\underline1}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {10:1--10:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.10}, URN = {urn:nbn:de:0030-drops-68456}, doi = {10.4230/LIPIcs.FSTTCS.2016.10}, annote = {Keywords: Metric Embeddings, Sparsest Cut, Negative type metrics, Approximation Algorithms} }

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

A recent result of Moshkovitz [Moshkovitz14] presented an ingenious method to provide a completely elementary proof of the Parallel Repetition Theorem for certain projection games via a construction called fortification. However, the construction used in [Moshkovitz14] to fortify arbitrary label cover instances using an arbitrary extractor is insufficient to prove parallel repetition. In this paper, we provide a fix by using a stronger graph that we call fortifiers. Fortifiers are graphs that have both l_1 and l_2 guarantees on induced distributions from large subsets.
We then show that an expander with sufficient spectral gap, or a bi-regular extractor with stronger parameters (the latter is also the construction used in an independent update [Moshkovitz15] of [Moshkovitz14] with an alternate argument), is a good fortifier. We also show that using a fortifier (in particular l_2 guarantees) is necessary for obtaining the robustness required for fortification.

Amey Bhangale, Ramprasad Saptharishi, Girish Varma, and Rakesh Venkat. On Fortification of Projection Games. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 497-511, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bhangale_et_al:LIPIcs.APPROX-RANDOM.2015.497, author = {Bhangale, Amey and Saptharishi, Ramprasad and Varma, Girish and Venkat, Rakesh}, title = {{On Fortification of Projection Games}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {497--511}, 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.2015.497}, URN = {urn:nbn:de:0030-drops-53204}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.497}, annote = {Keywords: Parallel Repetition, Fortification} }

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

Guruswami and Sinop give a O(1/delta) approximation guarantee for the non-uniform Sparsest Cut problem by solving O(r)-level Lasserre semidefinite constraints, provided that the generalized eigenvalues of the Laplacians of the cost and demand graphs satisfy a certain spectral condition, namely, the (r+1)-th generalized eigenvalue is at least OPT/(1-delta). Their key idea is a rounding technique that first maps a vector-valued solution to [0,1] using appropriately scaled projections onto Lasserre vectors. In this paper, we show that similar projections and analysis can be obtained using only l_2^2 triangle inequality constraints. This results in a O(r/delta^2) approximation guarantee for the non-uniform Sparsest Cut problem by adding only l_2^2 triangle inequality constraints to the usual semidefinite program, provided that the same spectral condition, the (r+1)-th generalized eigenvalue is at least OPT/(1-delta), holds.

Amit Deshpande and Rakesh Venkat. Guruswami-Sinop Rounding without Higher Level Lasserre. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 105-114, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{deshpande_et_al:LIPIcs.APPROX-RANDOM.2014.105, author = {Deshpande, Amit and Venkat, Rakesh}, title = {{Guruswami-Sinop Rounding without Higher Level Lasserre}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {105--114}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.105}, URN = {urn:nbn:de:0030-drops-46911}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.105}, annote = {Keywords: Sparsest Cut, Lasserre Hierarchy, Metric embeddings} }

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