15 Search Results for "Raghavendra, Prasad"


Document
Invited Talk
On Measuring Average Case Complexity via Sum-Of-Squares Degree (Invited Talk)

Authors: Prasad Raghavendra

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


Abstract
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.

Cite as

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}
}
Document
Track A: Algorithms, Complexity and Games
Approximating Max-Cut on Bounded Degree Graphs: Tighter Analysis of the FKL Algorithm

Authors: Jun-Ting Hsieh and Pravesh K. Kothari

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)


Abstract
In this note, we describe a α_GW + Ω̃(1/d²)-factor approximation algorithm for Max-Cut on weighted graphs of degree ⩽ d. Here, α_GW ≈ 0.878 is the worst-case approximation ratio of the Goemans-Williamson rounding for Max-Cut. This improves on previous results for unweighted graphs by Feige, Karpinski, and Langberg [Feige et al., 2002] and Florén [Florén, 2016]. Our guarantee is obtained by a tighter analysis of the solution obtained by applying a natural local improvement procedure to the Goemans-Williamson rounding of the basic SDP strengthened with triangle inequalities.

Cite as

Jun-Ting Hsieh and Pravesh K. Kothari. Approximating Max-Cut on Bounded Degree Graphs: Tighter Analysis of the FKL Algorithm. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 77:1-77:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{hsieh_et_al:LIPIcs.ICALP.2023.77,
  author =	{Hsieh, Jun-Ting and Kothari, Pravesh K.},
  title =	{{Approximating Max-Cut on Bounded Degree Graphs: Tighter Analysis of the FKL Algorithm}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{77:1--77:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.77},
  URN =		{urn:nbn:de:0030-drops-181291},
  doi =		{10.4230/LIPIcs.ICALP.2023.77},
  annote =	{Keywords: Max-Cut, approximation algorithm, semidefinite programming}
}
Document
Efficient Algorithms for Certifying Lower Bounds on the Discrepancy of Random Matrices

Authors: Prayaag Venkat

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


Abstract
In this paper, we initiate the study of the algorithmic problem of certifying lower bounds on the discrepancy of random matrices: given an input matrix A ∈ ℝ^{m × n}, output a value that is a lower bound on disc(A) = min_{x ∈ {± 1}ⁿ} ‖Ax‖_∞ for every A, but is close to the typical value of disc(A) with high probability over the choice of a random A. This problem is important because of its connections to conjecturally-hard average-case problems such as negatively-spiked PCA [Afonso S. Bandeira et al., 2020], the number-balancing problem [Gamarnik and Kızıldağ, 2021] and refuting random constraint satisfaction problems [Prasad Raghavendra et al., 2017]. We give the first polynomial-time algorithms with non-trivial guarantees for two main settings. First, when the entries of A are i.i.d. standard Gaussians, it is known that disc(A) = Θ (√n2^{-n/m}) with high probability [Karthekeyan Chandrasekaran and Santosh S. Vempala, 2014; Aubin et al., 2019; Paxton Turner et al., 2020] and that super-constant levels of the Sum-of-Squares SDP hierarchy fail to certify anything better than disc(A) ≥ 0 when m < n - o(n) [Mrinalkanti Ghosh et al., 2020]. In contrast, our algorithm certifies that disc(A) ≥ exp(-O(n²/m)) with high probability. As an application, this formally refutes a conjecture of Bandeira, Kunisky, and Wein [Afonso S. Bandeira et al., 2020] on the computational hardness of the detection problem in the negatively-spiked Wishart model. Second, we consider the integer partitioning problem: given n uniformly random b-bit integers a₁, …, a_n, certify the non-existence of a perfect partition, i.e. certify that disc(A) ≥ 1 for A = (a₁, …, a_n). Under the scaling b = α n, it is known that the probability of the existence of a perfect partition undergoes a phase transition from 1 to 0 at α = 1 [Christian Borgs et al., 2001]; our algorithm certifies the non-existence of perfect partitions for some α = O(n). We also give efficient non-deterministic algorithms with significantly improved guarantees, raising the possibility that the landscape of these certification problems closely resembles that of e.g. the problem of refuting random 3SAT formulas in the unsatisfiable regime. Our algorithms involve a reduction to the Shortest Vector Problem and employ the Lenstra-Lenstra-Lovász algorithm.

Cite as

Prayaag Venkat. Efficient Algorithms for Certifying Lower Bounds on the Discrepancy of Random Matrices. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 98:1-98:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{venkat:LIPIcs.ITCS.2023.98,
  author =	{Venkat, Prayaag},
  title =	{{Efficient Algorithms for Certifying Lower Bounds on the Discrepancy of Random Matrices}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{98:1--98:12},
  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.2023.98},
  URN =		{urn:nbn:de:0030-drops-176015},
  doi =		{10.4230/LIPIcs.ITCS.2023.98},
  annote =	{Keywords: Average-case discrepancy theory, lattices, shortest vector problem}
}
Document
Certifying Solution Geometry in Random CSPs: Counts, Clusters and Balance

Authors: Jun-Ting Hsieh, Sidhanth Mohanty, and Jeff Xu

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)


Abstract
An active topic in the study of random constraint satisfaction problems (CSPs) is the geometry of the space of satisfying or almost satisfying assignments as the function of the density, for which a precise landscape of predictions has been made via statistical physics-based heuristics. In parallel, there has been a recent flurry of work on refuting random constraint satisfaction problems, via nailing refutation thresholds for spectral and semidefinite programming-based algorithms, and also on counting solutions to CSPs. Inspired by this, the starting point for our work is the following question: What does the solution space for a random CSP look like to an efficient algorithm? In pursuit of this inquiry, we focus on the following problems about random Boolean CSPs at the densities where they are unsatisfiable but no refutation algorithm is known. 1) Counts. For every Boolean CSP we give algorithms that with high probability certify a subexponential upper bound on the number of solutions. We also give algorithms to certify a bound on the number of large cuts in a Gaussian-weighted graph, and the number of large independent sets in a random d-regular graph. 2) Clusters. For Boolean 3CSPs we give algorithms that with high probability certify an upper bound on the number of clusters of solutions. 3) Balance. We also give algorithms that with high probability certify that there are no "unbalanced" solutions, i.e., solutions where the fraction of +1s deviates significantly from 50%. Finally, we also provide hardness evidence suggesting that our algorithms for counting are optimal.

Cite as

Jun-Ting Hsieh, Sidhanth Mohanty, and Jeff Xu. Certifying Solution Geometry in Random CSPs: Counts, Clusters and Balance. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{hsieh_et_al:LIPIcs.CCC.2022.11,
  author =	{Hsieh, Jun-Ting and Mohanty, Sidhanth and Xu, Jeff},
  title =	{{Certifying Solution Geometry in Random CSPs: Counts, Clusters and Balance}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{11:1--11:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.11},
  URN =		{urn:nbn:de:0030-drops-165735},
  doi =		{10.4230/LIPIcs.CCC.2022.11},
  annote =	{Keywords: constraint satisfaction problems, certified counting, random graphs}
}
Document
The SDP Value for Random Two-Eigenvalue CSPs

Authors: Sidhanth Mohanty, Ryan O'Donnell, and Pedro Paredes

Published in: LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)


Abstract
We precisely determine the SDP value (equivalently, quantum value) of large random instances of certain kinds of constraint satisfaction problems, "two-eigenvalue 2CSPs". We show this SDP value coincides with the spectral relaxation value, possibly indicating a computational threshold. Our analysis extends the previously resolved cases of random regular 2XOR and NAE-3SAT, and includes new cases such as random Sort₄ (equivalently, CHSH) and Forrelation CSPs. Our techniques include new generalizations of the nonbacktracking operator, the Ihara-Bass Formula, and the Friedman/Bordenave proof of Alon’s Conjecture.

Cite as

Sidhanth Mohanty, Ryan O'Donnell, and Pedro Paredes. The SDP Value for Random Two-Eigenvalue CSPs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 50:1-50:45, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{mohanty_et_al:LIPIcs.STACS.2020.50,
  author =	{Mohanty, Sidhanth and O'Donnell, Ryan and Paredes, Pedro},
  title =	{{The SDP Value for Random Two-Eigenvalue CSPs}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{50:1--50:45},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.50},
  URN =		{urn:nbn:de:0030-drops-119110},
  doi =		{10.4230/LIPIcs.STACS.2020.50},
  annote =	{Keywords: Semidefinite programming, constraint satisfaction problems}
}
Document
A Note on Max k-Vertex Cover: Faster FPT-AS, Smaller Approximate Kernel and Improved Approximation

Authors: Pasin Manurangsi

Published in: OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)


Abstract
In Maximum k-Vertex Cover (Max k-VC), the input is an edge-weighted graph G and an integer k, and the goal is to find a subset S of k vertices that maximizes the total weight of edges covered by S. Here we say that an edge is covered by S iff at least one of its endpoints lies in S. We present an FPT approximation scheme (FPT-AS) that runs in (1/epsilon)^{O(k)} poly(n) time for the problem, which improves upon Gupta, Lee and Li's (k/epsilon)^{O(k)} poly(n)-time FPT-AS [Anupam Gupta and, 2018; Anupam Gupta et al., 2018]. Our algorithm is simple: just use brute force to find the best k-vertex subset among the O(k/epsilon) vertices with maximum weighted degrees. Our algorithm naturally yields an (efficient) approximate kernelization scheme of O(k/epsilon) vertices; previously, an O(k^5/epsilon^2)-vertex approximate kernel is only known for the unweighted version of Max k-VC [Daniel Lokshtanov and, 2017]. Interestingly, this also has an application outside of parameterized complexity: using our approximate kernelization as a preprocessing step, we can directly apply Raghavendra and Tan's SDP-based algorithm for 2SAT with cardinality constraint [Prasad Raghavendra and, 2012] to give an 0.92-approximation algorithm for Max k-VC in polynomial time. This improves upon the best known polynomial time approximation algorithm of Feige and Langberg [Uriel Feige and, 2001] which yields (0.75 + delta)-approximation for some (small and unspecified) constant delta > 0. We also consider the minimization version of the problem (called Min k-VC), where the goal is to find a set S of k vertices that minimizes the total weight of edges covered by S. We provide a FPT-AS for Min k-VC with similar running time of (1/epsilon)^{O(k)} poly(n). Once again, this improves on a (k/epsilon)^{O(k)} poly(n)-time FPT-AS of Gupta et al. On the other hand, we show, assuming a variant of the Small Set Expansion Hypothesis [Raghavendra and Steurer, 2010] and NP !subseteq coNP/poly, that there is no polynomial size approximate kernelization for Min k-VC for any factor less than two.

Cite as

Pasin Manurangsi. A Note on Max k-Vertex Cover: Faster FPT-AS, Smaller Approximate Kernel and Improved Approximation. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 15:1-15:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{manurangsi:OASIcs.SOSA.2019.15,
  author =	{Manurangsi, Pasin},
  title =	{{A Note on Max k-Vertex Cover: Faster FPT-AS, Smaller Approximate Kernel and Improved Approximation}},
  booktitle =	{2nd Symposium on Simplicity in Algorithms (SOSA 2019)},
  pages =	{15:1--15:21},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-099-6},
  ISSN =	{2190-6807},
  year =	{2019},
  volume =	{69},
  editor =	{Fineman, Jeremy T. and Mitzenmacher, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.15},
  URN =		{urn:nbn:de:0030-drops-100417},
  doi =		{10.4230/OASIcs.SOSA.2019.15},
  annote =	{Keywords: Maximum k-Vertex Cover, Minimum k-Vertex Cover, Approximation Algorithms, Fixed Parameter Algorithms, Approximate Kernelization}
}
Document
Average Whenever You Meet: Opportunistic Protocols for Community Detection

Authors: Luca Becchetti, Andrea Clementi, Pasin Manurangsi, Emanuele Natale, Francesco Pasquale, Prasad Raghavendra, and Luca Trevisan

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


Abstract
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.

Cite as

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}
}
Document
Dimension Reduction for Polynomials over Gaussian Space and Applications

Authors: Badih Ghazi, Pritish Kamath, and Prasad Raghavendra

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


Abstract
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.

Cite as

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}
}
Document
Real Stability Testing

Authors: Prasad Raghavendra, Nick Ryder, and Nikhil Srivastava

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


Abstract
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.

Cite as

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}
}
Document
A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs

Authors: Pasin Manurangsi and Prasad Raghavendra

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


Abstract
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.

Cite as

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}
}
Document
On the Bit Complexity of Sum-of-Squares Proofs

Authors: Prasad Raghavendra and Benjamin Weitz

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


Abstract
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).

Cite as

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}
}
Document
Correlation Decay and Tractability of CSPs

Authors: Jonah Brown-Cohen and Prasad Raghavendra

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


Abstract
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.

Cite as

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}
}
Document
Beating the Random Assignment on Constraint Satisfaction Problems of Bounded Degree

Authors: Boaz Barak, Ankur Moitra, Ryan O’Donnell, Prasad Raghavendra, Oded Regev, David Steurer, Luca Trevisan, Aravindan Vijayaraghavan, David Witmer, and John Wright

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


Abstract
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.

Cite as

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}
}
Document
Gap Amplification for Small-Set Expansion via Random Walks

Authors: Prasad Raghavendra and Tselil Schramm

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


Abstract
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.

Cite as

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}
}
Document
Primal Infon Logic: Derivability in Polynomial Time

Authors: Anguraj Baskar, Prasad Naldurg, K. R. Raghavendra, and S. P. Suresh

Published in: LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)


Abstract
Primal infon logic (PIL), introduced by Gurevich and Neeman in 2009, is a logic for authorization in distributed systems. It is a variant of the (and, implies)-fragment of intuitionistic modal logic. It presents many interesting technical challenges -- one of them is to determine the complexity of the derivability problem. Previously, some restrictions of propositional PIL were proved to have a linear time algorithm, and some extensions have been proved to be PSPACE-complete. In this paper, we provide an O(N^3) algorithm for derivability in propositional PIL. The solution involves an interesting interplay between the sequent calculus formulation (to prove the subformula property) and the natural deduction formulation of the logic (based on which we provide an algorithm for the derivability problem).

Cite as

Anguraj Baskar, Prasad Naldurg, K. R. Raghavendra, and S. P. Suresh. Primal Infon Logic: Derivability in Polynomial Time. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 163-174, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


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@InProceedings{baskar_et_al:LIPIcs.FSTTCS.2013.163,
  author =	{Baskar, Anguraj and Naldurg, Prasad and Raghavendra, K. R. and Suresh, S. P.},
  title =	{{Primal Infon Logic: Derivability in Polynomial Time}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)},
  pages =	{163--174},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-64-4},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{24},
  editor =	{Seth, Anil and Vishnoi, Nisheeth K.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.163},
  URN =		{urn:nbn:de:0030-drops-43708},
  doi =		{10.4230/LIPIcs.FSTTCS.2013.163},
  annote =	{Keywords: Authorization logics, Intuitionistic modal logic, Proof theory, Cut elimination, Subformula property}
}
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