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Invited Talk

**Published in:** LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

Sum-of-squares semidefinite programming hierarchy is a sequence of increasingly complex semidefinite programs to reason about systems of polynomial inequalities. The k-th-level of the sum-of-squares SDP hierarchy is a semidefinite program that can be solved in time n^O(k).
Sum-of-squares SDP hierarchies subsume fundamental algorithmic techniques such as linear programming and spectral methods. Many state-of-the-art algorithms for approximating NP-hard optimization problems are captured in the first few levels of the hierarchy. More recently, sum-of-squares SDPs have been applied extensively towards designing algorithms for average case problems. These include planted problems, random constraint satisfaction problems, and computational problems arising in statistics.
From the standpoint of complexity theory, sum-of-squares SDPs can be applied towards measuring the average-case hardness of a problem. Most natural optimization problems can often be shown to be solvable by degree n sum-of-squares SDP, which corresponds to an exponential time algorithm. The smallest degree of the sum-of-squares relaxation needed to solve a problem can be used as a measure of the computational complexity of the problem. This approach seems especially useful for understanding average-case complexity under natural distributions. For example, the sum-of-squares degree has been used to nearly characterize the computational complexity of refuting random CSPs as a function of the number of constraints.
Using the sum-of-squares degree as a proxy measure for average case complexity opens the door to formalizing certain computational phase transitions that have been conjectured for average case problems such as recovery in stochastic block models.
In this talk, we discuss applications of this approach to average-case complexity and present some open problems.

Prasad Raghavendra. On Measuring Average Case Complexity via Sum-Of-Squares Degree (Invited Talk). In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{raghavendra:LIPIcs.FSTTCS.2023.2, author = {Raghavendra, Prasad}, title = {{On Measuring Average Case Complexity via Sum-Of-Squares Degree}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {2:1--2:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-304-1}, ISSN = {1868-8969}, year = {2023}, volume = {284}, editor = {Bouyer, Patricia and Srinivasan, Srikanth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.2}, URN = {urn:nbn:de:0030-drops-193750}, doi = {10.4230/LIPIcs.FSTTCS.2023.2}, annote = {Keywords: semidefinite programming, sum-of-squares SDP, average case complexity, random SAT, stochastic block models} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Consider the following asynchronous, opportunistic communication model over a graph G: in each round, one edge is activated uniformly and independently at random and (only) its two endpoints can exchange messages and perform local computations. Under this model, we study the following random process: The first time a vertex is an endpoint of an active edge, it chooses a random number, say +/- 1 with probability 1/2; then, in each round, the two endpoints of the currently active edge update their values to their average.
We provide a rigorous analysis of the above process showing that, if G exhibits a two-community structure (for example, two expanders connected by a sparse cut), the values held by the nodes will collectively reflect the underlying community structure over a suitable phase of the above process. Our analysis requires new concentration bounds on the product of certain random matrices that are technically challenging and possibly of independent interest.
We then exploit our analysis to design the first opportunistic protocols that approximately recover community structure using only logarithmic (or polylogarithmic, depending on the sparsity of the cut) work per node.

Luca Becchetti, Andrea Clementi, Pasin Manurangsi, Emanuele Natale, Francesco Pasquale, Prasad Raghavendra, and Luca Trevisan. Average Whenever You Meet: Opportunistic Protocols for Community Detection. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{becchetti_et_al:LIPIcs.ESA.2018.7, author = {Becchetti, Luca and Clementi, Andrea and Manurangsi, Pasin and Natale, Emanuele and Pasquale, Francesco and Raghavendra, Prasad and Trevisan, Luca}, title = {{Average Whenever You Meet: Opportunistic Protocols for Community Detection}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {7:1--7:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah 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.2018.7}, URN = {urn:nbn:de:0030-drops-94705}, doi = {10.4230/LIPIcs.ESA.2018.7}, annote = {Keywords: Community Detection, Random Processes, Spectral Analysis} }

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**Published in:** LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

We introduce a new technique for reducing the dimension of the ambient space of low-degree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilon-optimal noise-stable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilon-approximate any joint distribution that can be non-interactively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermann-like to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap non-interactive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016].
Our technique of dimension reduction for low-degree polynomials is simple and can be seen as a generalization of the Johnson-Lindenstrauss lemma and could be of independent interest.

Badih Ghazi, Pritish Kamath, and Prasad Raghavendra. Dimension Reduction for Polynomials over Gaussian Space and Applications. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 28:1-28:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ghazi_et_al:LIPIcs.CCC.2018.28, author = {Ghazi, Badih and Kamath, Pritish and Raghavendra, Prasad}, title = {{Dimension Reduction for Polynomials over Gaussian Space and Applications}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {28:1--28:37}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.28}, URN = {urn:nbn:de:0030-drops-88616}, doi = {10.4230/LIPIcs.CCC.2018.28}, annote = {Keywords: Dimension reduction, Low-degree Polynomials, Noise Stability, Non-Interactive Simulation} }

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

We give a strongly polynomial time algorithm which determines whether or not a bivariate polynomial is real stable. As a corollary, this implies an algorithm for testing whether a given linear transformation on univariate polynomials preserves real-rootedness. The proof exploits properties of hyperbolic polynomials to reduce real stability testing to testing nonnegativity of a finite number of polynomials on an interval.

Prasad Raghavendra, Nick Ryder, and Nikhil Srivastava. Real Stability Testing. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{raghavendra_et_al:LIPIcs.ITCS.2017.5, author = {Raghavendra, Prasad and Ryder, Nick and Srivastava, Nikhil}, title = {{Real Stability Testing}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {5:1--5:15}, 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.2017.5}, URN = {urn:nbn:de:0030-drops-81965}, doi = {10.4230/LIPIcs.ITCS.2017.5}, annote = {Keywords: real stable polynomials, hyperbolic polynomials, real rootedness, moment matrix, sturm sequence} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

A (k x l)-birthday repetition G^{k x l} of a two-prover game G is a game in which the two provers are sent random sets of questions from G of sizes k and l respectively. These two sets are sampled independently uniformly among all sets of questions of those particular sizes. We prove the following birthday repetition theorem: when G satisfies some mild conditions, val(G^{k x l}) decreases exponentially in Omega(kl/n) where n is the total number of questions. Our result positively resolves an open question posted by Aaronson, Impagliazzo and Moshkovitz [Aaronson et al., CCC, 2014].
As an application of our birthday repetition theorem, we obtain new fine-grained inapproximability results for dense CSPs. Specifically, we establish a tight trade-off between running time and approximation ratio by showing conditional lower bounds, integrality gaps and approximation algorithms; in particular, for any sufficiently large i and for every k >= 2, we show the following:
- We exhibit an O(q^{1/i})-approximation algorithm for dense Max k-CSPs with alphabet size q via O_k(i)-level of Sherali-Adams relaxation.
- Through our birthday repetition theorem, we obtain an integrality gap of q^{1/i} for Omega_k(i / polylog i)-level Lasserre relaxation for fully-dense Max k-CSP.
- Assuming that there is a constant epsilon > 0 such that Max 3SAT cannot be approximated to within (1 - epsilon) of the optimal in sub-exponential time, our birthday repetition theorem implies that any algorithm that approximates fully-dense Max k-CSP to within a q^{1/i} factor takes (nq)^{Omega_k(i / polylog i)} time, almost tightly matching our algorithmic result.
As a corollary of our algorithm for dense Max k-CSP, we give a new approximation algorithm for Densest k-Subhypergraph, a generalization of Densest k-Subgraph to hypergraphs. When the input hypergraph is O(1)-uniform and the optimal k-subhypergraph has constant density, our algorithm finds a k-subhypergraph of density Omega(n^{−1/i}) in time n^{O(i)} for any integer i > 0.

Pasin Manurangsi and Prasad Raghavendra. A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 78:1-78:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{manurangsi_et_al:LIPIcs.ICALP.2017.78, author = {Manurangsi, Pasin and Raghavendra, Prasad}, title = {{A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {78:1--78:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.78}, URN = {urn:nbn:de:0030-drops-74638}, doi = {10.4230/LIPIcs.ICALP.2017.78}, annote = {Keywords: Birthday Repetition, Constraint Satisfaction Problems, Linear Program} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

It has often been claimed in recent papers that one can find a degree d Sum-of-Squares proof if one exists via the Ellipsoid algorithm. In a recent paper, Ryan O'Donnell notes this widely quoted claim is not necessarily true. He presents an example of a polynomial system with bounded coefficients that admits low-degree proofs of non-negativity, but these proofs necessarily involve numbers with an exponential number of bits, causing the Ellipsoid algorithm to take exponential time. In this paper we obtain both positive and negative results on the bit complexity of SoS proofs.
First, we propose a sufficient condition on a polynomial system that implies a bound on the coefficients in an SoS proof. We demonstrate that this sufficient condition is applicable for common use-cases of the SoS algorithm, such as Max-CSP, Balanced Separator, Max-Clique, Max-Bisection, and Unit-Vector constraints.
On the negative side, O'Donnell asked whether every polynomial system containing Boolean constraints admits proofs of polynomial bit complexity. We answer this question in the negative, giving a counterexample system and non-negative polynomial which has degree two SoS proofs, but no SoS proof with small coefficients until degree sqrt(n).

Prasad Raghavendra and Benjamin Weitz. On the Bit Complexity of Sum-of-Squares Proofs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 80:1-80:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{raghavendra_et_al:LIPIcs.ICALP.2017.80, author = {Raghavendra, Prasad and Weitz, Benjamin}, title = {{On the Bit Complexity of Sum-of-Squares Proofs}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {80:1--80:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.80}, URN = {urn:nbn:de:0030-drops-73804}, doi = {10.4230/LIPIcs.ICALP.2017.80}, annote = {Keywords: Sum-of-Squares, Combinatorial Optimization, Proof Complexity} }

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

The algebraic dichotomy conjecture of Bulatov, Krokhin and Jeavons yields an elegant characterization of the complexity of constraint satisfaction problems. Roughly speaking, the characterization asserts that a CSP L is tractable if and only if there exist certain non-trivial operations known as polymorphisms to combine solutions to L to create new ones.
In this work, we study the dynamical system associated with repeated applications of a polymorphism to a distribution over assignments. Specifically, we exhibit a correlation decay phenomenon that makes two variables or groups of variables that are not perfectly correlated become independent after repeated applications of a polymorphism.
We show that this correlation decay phenomenon can be utilized in designing algorithms for CSPs by exhibiting two applications:
1. A simple randomized algorithm to solve linear equations over a prime field, whose analysis crucially relies on correlation decay.
2. A sufficient condition for the simple linear programming relaxation for a 2-CSP to be sound (have no integrality gap) on a given instance.

Jonah Brown-Cohen and Prasad Raghavendra. Correlation Decay and Tractability of CSPs. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 79:1-79:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{browncohen_et_al:LIPIcs.ICALP.2016.79, author = {Brown-Cohen, Jonah and Raghavendra, Prasad}, title = {{Correlation Decay and Tractability of CSPs}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {79:1--79:13}, 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.2016.79}, URN = {urn:nbn:de:0030-drops-62064}, doi = {10.4230/LIPIcs.ICALP.2016.79}, annote = {Keywords: Constraint Satisfaction, Polymorphisms, Linear Equations, Correlation Decay} }

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

We show that for any odd k and any instance I of the max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a 1/2 + Omega(1/sqrt(D)) fraction of I's constraints, where D is a bound on the number of constraints that each variable occurs in.
This improves both qualitatively and quantitatively on the recent work of Farhi, Goldstone, and Gutmann (2014), which gave a quantum algorithm to find an assignment satisfying a 1/2 Omega(D^{-3/4}) fraction of the equations.
For arbitrary constraint satisfaction problems, we give a similar result for "triangle-free" instances; i.e., an efficient algorithm that finds an assignment satisfying at least a mu + Omega(1/sqrt(degree)) fraction of constraints, where mu is the fraction that would be satisfied by a uniformly random assignment.

Boaz Barak, Ankur Moitra, Ryan O’Donnell, Prasad Raghavendra, Oded Regev, David Steurer, Luca Trevisan, Aravindan Vijayaraghavan, David Witmer, and John Wright. Beating the Random Assignment on Constraint Satisfaction Problems of Bounded Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 110-123, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{barak_et_al:LIPIcs.APPROX-RANDOM.2015.110, author = {Barak, Boaz and Moitra, Ankur and O’Donnell, Ryan and Raghavendra, Prasad and Regev, Oded and Steurer, David and Trevisan, Luca and Vijayaraghavan, Aravindan and Witmer, David and Wright, John}, title = {{Beating the Random Assignment on Constraint Satisfaction Problems of Bounded Degree}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {110--123}, 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.2015.110}, URN = {urn:nbn:de:0030-drops-52981}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.110}, annote = {Keywords: constraint satisfaction problems, bounded degree, advantage over random} }

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

In this work, we achieve gap amplification for the Small-Set Expansion problem. Specifically, we show that an instance of the Small-Set Expansion Problem with completeness epsilon and soundness 1/2 is at least as difficult as Small-Set Expansion with completeness epsilon and soundness f(epsilon), for any function f(epsilon) which grows faster than (epsilon)^(1/2). We achieve this amplification via random walks--the output graph corresponds to taking random walks on the original graph. An interesting feature of our reduction is that unlike gap amplification via parallel repetition, the size of the instances (number of vertices) produced by the reduction remains the same.

Prasad Raghavendra and Tselil Schramm. Gap Amplification for Small-Set Expansion via Random Walks. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 381-391, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{raghavendra_et_al:LIPIcs.APPROX-RANDOM.2014.381, author = {Raghavendra, Prasad and Schramm, Tselil}, title = {{Gap Amplification for Small-Set Expansion via Random Walks}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {381--391}, 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.2014.381}, URN = {urn:nbn:de:0030-drops-47108}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.381}, annote = {Keywords: Gap amplification, Small-Set Expansion, random walks, graph products, Unique Games} }

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