All reports by Author Venkatesan Guruswami:

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TR23-126
| 25th August 2023
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Omar Alrabiah, Venkatesan Guruswami, Ray Li#### AG codes have no list-decoding friends: Approaching the generalized Singleton bound requires exponential alphabets

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TR23-125
| 25th August 2023
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Omar Alrabiah, Venkatesan Guruswami, Ray Li#### Randomly punctured Reed-Solomon codes achieve list-decoding capacity over linear-sized fields

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TR21-145
| 19th October 2021
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Omar Alrabiah, Venkatesan Guruswami#### Revisiting a Lower Bound on the Redundancy of Linear Batch Codes

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TR21-139
| 24th September 2021
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Venkatesan Guruswami, Jonathan Mosheiff#### Punctured Large Distance Codes, and Many Reed-Solomon Codes, Achieve List-Decoding Capacity

Revisions: 2

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TR21-119
| 13th August 2021
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Omar Alrabiah, Venkatesan Guruswami#### Visible Rank and Codes with Locality

Revisions: 2

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TR21-112
| 30th July 2021
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Vikraman Arvind, Venkatesan Guruswami#### CNF Satisfiability in a Subspace and Related Problems

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TR21-079
| 9th June 2021
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Venkatesan Guruswami, Xiaoyu He, Ray Li#### The zero-rate threshold for adversarial bit-deletions is less than 1/2

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TR21-026
| 23rd February 2021
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Joshua Brakensiek, Venkatesan Guruswami, Sai Sandeep#### Conditional Dichotomy of Boolean Ordered Promise CSPs

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TR21-025
| 15th February 2021
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Sivakanth Gopi, Venkatesan Guruswami#### Improved Maximally Recoverable LRCs using Skew Polynomials

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TR20-172
| 13th November 2020
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Venkatesan Guruswami, Chaoping Xing#### Optimal rate list decoding over bounded alphabets using algebraic-geometric codes

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TR20-167
| 9th November 2020
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Venkatesan Guruswami, Sai Sandeep#### Approximate Hypergraph Vertex Cover and generalized Tuza's conjecture

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TR20-151
| 8th October 2020
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Venkatesan Guruswami, Vinayak Kumar#### Pseudobinomiality of the Sticky Random Walk

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TR20-004
| 17th January 2020
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Joshua Brakensiek, Venkatesan Guruswami, Marcin Wrochna, Stanislav Zivny#### The Power of the Combined Basic LP and Affine Relaxation for Promise CSPs

Revisions: 1

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TR19-154
| 6th November 2019
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Venkatesan Guruswami, Andrii Riazanov, Min Ye#### Ar?kan meets Shannon: Polar codes with near-optimal convergence to channel capacity

Revisions: 3

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TR19-153
| 6th November 2019
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Venkatesan Guruswami, Bernhard Haeupler, Amirbehshad Shahrasbi#### Optimally Resilient Codes for List-Decoding from Insertions and Deletions

Revisions: 1

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TR19-116
| 9th September 2019
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Venkatesan Guruswami, Sai Sandeep#### $d$-to-$1$ Hardness of Coloring $4$-colorable Graphs with $O(1)$ colors

Revisions: 1

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TR19-094
| 16th July 2019
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Venkatesan Guruswami, Sai Sandeep#### Rainbow coloring hardness via low sensitivity polymorphisms

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TR19-092
| 9th July 2019
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Venkatesan Guruswami, Jakub Opršal, Sai Sandeep#### Revisiting Alphabet Reduction in Dinur's PCP

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TR19-054
| 9th April 2019
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Joshua Brakensiek, Venkatesan Guruswami#### Bridging between 0/1 and Linear Programming via Random Walks

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TR19-013
| 31st January 2019
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Joshua Brakensiek, Sivakanth Gopi, Venkatesan Guruswami#### CSPs with Global Modular Constraints: Algorithms and Hardness via Polynomial Representations

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TR19-005
| 16th January 2019
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Omar Alrabiah, Venkatesan Guruswami#### An Exponential Lower Bound on the Sub-Packetization of MSR Codes

Revisions: 1

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TR18-097
| 15th May 2018
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Vijay Bhattiprolu, Mrinalkanti Ghosh, Venkatesan Guruswami, Euiwoong Lee, Madhur Tulsiani#### Approximating Operator Norms via Generalized Krivine Rounding

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TR18-096
| 13th May 2018
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Venkatesan Guruswami, Andrii Riazanov#### Beating Fredman-Komlós for perfect $k$-hashing

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TR18-059
| 6th April 2018
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Joshua Brakensiek, Venkatesan Guruswami#### Combining LPs and Ring Equations via Structured Polymorphisms

Revisions: 1

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TR18-037
| 21st February 2018
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Vijay Bhattiprolu, Mrinalkanti Ghosh, Venkatesan Guruswami, Euiwoong Lee, Madhur Tulsiani#### Inapproximability of Matrix $p \rightarrow q$ Norms

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TR18-027
| 8th February 2018
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Jaroslaw Blasiok, Venkatesan Guruswami, Preetum Nakkiran, Atri Rudra, Madhu Sudan#### General Strong Polarization

Revisions: 1

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TR18-017
| 26th January 2018
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Venkatesan Guruswami, Nicolas Resch, Chaoping Xing#### Lossless dimension expanders via linearized polynomials and subspace designs

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TR17-183
| 28th November 2017
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Sivakanth Gopi, Venkatesan Guruswami, Sergey Yekhanin#### On Maximally Recoverable Local Reconstruction Codes

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TR17-147
| 3rd October 2017
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Venkatesan Guruswami, Rishi Saket#### Hardness of Rainbow Coloring Hypergraphs

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TR17-141
| 19th September 2017
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Joshua Brakensiek, Venkatesan Guruswami#### A Family of Dictatorship Tests with Perfect Completeness for 2-to-2 Label Cover

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TR17-080
| 1st May 2017
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Joshua Brakensiek, Venkatesan Guruswami#### The Quest for Strong Inapproximability Results with Perfect Completeness

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TR17-064
| 20th April 2017
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Venkatesan Guruswami, Chaoping Xing, chen yuan#### Subspace Designs based on Algebraic Function Fields

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TR16-185
| 18th November 2016
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Vijay Bhattiprolu, Mrinalkanti Ghosh, Venkatesan Guruswami, Euiwoong Lee, Madhur Tulsiani#### Multiplicative Approximations for Polynomial Optimization Over the Unit Sphere

Revisions: 1

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TR16-183
| 16th November 2016
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Joshua Brakensiek, Venkatesan Guruswami#### Promise Constraint Satisfaction: Algebraic Structure and a Symmetric Boolean Dichotomy

Revisions: 2

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TR16-033
| 10th March 2016
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Venkatesan Guruswami, Jaikumar Radhakrishnan#### Tight bounds for communication assisted agreement distillation

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TR16-029
| 7th March 2016
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Joshua Brakensiek, Venkatesan Guruswami#### New hardness results for graph and hypergraph colorings

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TR15-155
| 22nd September 2015
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Venkatesan Guruswami, Euiwoong Lee#### Nearly Optimal NP-Hardness of Unique Coverage

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TR15-117
| 21st July 2015
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Boris Bukh, Venkatesan Guruswami#### An improved bound on the fraction of correctable deletions

Revisions: 1

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TR15-116
| 21st July 2015
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Joshua Brakensiek, Venkatesan Guruswami, Samuel Zbarsky#### Efficient Low-Redundancy Codes for Correcting Multiple Deletions

Revisions: 1

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TR15-105
| 21st June 2015
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Venkatesan Guruswami, Euiwoong Lee#### Towards a Characterization of Approximation Resistance for Symmetric CSPs

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TR14-165
| 3rd December 2014
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Venkatesan Guruswami, Ameya Velingker#### An Entropy Sumset Inequality and Polynomially Fast Convergence to Shannon Capacity Over All Alphabets

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TR14-162
| 28th November 2014
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Michael Forbes, Venkatesan Guruswami#### Dimension Expanders via Rank Condensers

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TR14-153
| 14th November 2014
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Clement Canonne, Venkatesan Guruswami, Raghu Meka, Madhu Sudan#### Communication with Imperfectly Shared Randomness

Revisions: 3

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TR14-067
| 4th May 2014
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Venkatesan Guruswami, Madhu Sudan, Ameya Velingker, Carol Wang#### Limitations on Testable Affine-Invariant Codes in the High-Rate Regime

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TR14-043
| 2nd April 2014
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Venkatesan Guruswami, Euiwoong Lee#### Strong Inapproximability Results on Balanced Rainbow-Colorable Hypergraphs

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TR14-006
| 16th January 2014
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Venkatesan Guruswami, Euiwoong Lee#### Inapproximability of Feedback Vertex Set for Bounded Length Cycles

Revisions: 1

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TR13-175
| 6th December 2013
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Venkatesan Guruswami, Chaoping Xing#### Hitting Sets for Low-Degree Polynomials with Optimal Density

Revisions: 1

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TR13-170
| 2nd December 2013
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Venkatesan Guruswami, Carol Wang#### Explicit rank-metric codes list-decodable with optimal redundancy

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TR13-167
| 28th November 2013
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Venkatesan Guruswami, Prahladh Harsha, Johan Håstad, Srikanth Srinivasan, Girish Varma#### Super-polylogarithmic hypergraph coloring hardness via low-degree long codes

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TR13-159
| 20th November 2013
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Per Austrin, Venkatesan Guruswami, Johan Håstad#### $(2+\epsilon)$-SAT is NP-hard

Revisions: 2

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TR13-125
| 11th September 2013
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Venkatesan Guruswami, Euiwoong Lee#### Complexity of approximating CSP with Balance / Hard Constraints

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TR13-122
| 5th September 2013
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Irit Dinur, Venkatesan Guruswami#### PCPs via low-degree long code and hardness for constrained hypergraph coloring

Revisions: 1

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TR13-121
| 4th September 2013
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Mahdi Cheraghchi, Venkatesan Guruswami#### Non-Malleable Coding Against Bit-wise and Split-State Tampering

Revisions: 1

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TR13-118
| 2nd September 2013
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Mahdi Cheraghchi, Venkatesan Guruswami#### Capacity of Non-Malleable Codes

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TR13-071
| 8th May 2013
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Venkatesan Guruswami, Sushant Sachdeva, Rishi Saket#### Inapproximability of Minimum Vertex Cover on $k$-uniform $k$-partite Hypergraphs

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TR13-060
| 10th April 2013
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Venkatesan Guruswami, Swastik Kopparty#### Explicit Subspace Designs

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TR13-050
| 1st April 2013
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Venkatesan Guruswami, Patrick Xia#### Polar Codes: Speed of polarization and polynomial gap to capacity

Revisions: 1

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TR13-046
| 27th March 2013
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Venkatesan Guruswami, Chaoping Xing#### Optimal rate list decoding of folded algebraic-geometric codes over constant-sized alphabets

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TR13-002
| 31st December 2012
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Venkatesan Guruswami, Krzysztof Onak#### Superlinear lower bounds for multipass graph processing

Revisions: 3

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TR12-146
| 7th November 2012
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Venkatesan Guruswami, Chaoping Xing#### List decoding Reed-Solomon, Algebraic-Geometric, and Gabidulin subcodes up to the Singleton bound

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TR12-111
| 5th September 2012
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Venkatesan Guruswami, Ali Kemal Sinop#### Faster SDP hierarchy solvers for local rounding algorithms

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TR12-082
| 28th June 2012
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Mahdi Cheraghchi, Venkatesan Guruswami, Ameya Velingker#### Restricted Isometry of Fourier Matrices and List Decodability of Random Linear Codes

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TR12-074
| 12th June 2012
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Venkatesan Guruswami, Yuan Zhou#### Approximating Bounded Occurrence Ordering CSPs

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TR12-073
| 11th June 2012
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Venkatesan Guruswami, Carol Wang#### Linear-algebraic list decoding for variants of Reed-Solomon codes

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TR12-036
| 12th April 2012
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Venkatesan Guruswami, Chaoping Xing#### Folded Codes from Function Field Towers and Improved Optimal Rate List Decoding

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TR12-017
| 1st March 2012
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Venkatesan Guruswami, Srivatsan Narayanan#### Combinatorial limitations of a strong form of list decoding

Revisions: 1

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TR11-066
| 25th April 2011
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Venkatesan Guruswami, Ali Kemal Sinop#### Lasserre Hierarchy, Higher Eigenvalues, and Approximation Schemes for Quadratic Integer Programming with PSD Objectives

Revisions: 1

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TR11-027
| 28th February 2011
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Venkatesan Guruswami, Johan Håstad, Rajsekar Manokaran, Prasad Raghavendra, Moses Charikar#### Beating the Random Ordering is Hard: Every ordering CSP is approximation resistant

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TR10-185
| 2nd December 2010
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Vitaly Feldman, Venkatesan Guruswami, Prasad Raghavendra, Yi Wu#### Agnostic Learning of Monomials by Halfspaces is Hard

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TR10-177
| 16th November 2010
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Venkatesan Guruswami, Prasad Raghavendra, Rishi Saket, Yi Wu#### Bypassing UGC from some optimal geometric inapproximability results

Revisions: 1

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TR10-111
| 14th July 2010
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Venkatesan Guruswami, Ali Kemal Sinop#### The complexity of finding independent sets in bounded degree (hyper)graphs of low chromatic number

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TR10-077
| 26th April 2010
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Venkatesan Guruswami, Adam Smith#### Codes for Computationally Simple Channels: Explicit Constructions with Optimal Rate

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TR10-063
| 12th April 2010
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Venkatesan Guruswami, Yuan Zhou#### Tight Bounds on the Approximability of Almost-satisfiable Horn SAT and Exact Hitting Set}

Revisions: 1

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TR10-003
| 6th January 2010
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Venkatesan Guruswami, Johan Håstad, Swastik Kopparty#### On the List-Decodability of Random Linear Codes

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TR09-126
| 26th November 2009
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Eli Ben-Sasson, Venkatesan Guruswami, Tali Kaufman, Madhu Sudan, Michael Viderman#### Locally Testable Codes Require Redundant Testers

Revisions: 3

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TR09-099
| 16th October 2009
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Venkatesan Guruswami, Ali Kemal Sinop#### Improved Inapproximability Results for Maximum k-Colorable Subgraph

Revisions: 1

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TR09-020
| 2nd March 2009
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Venkatesan Guruswami, Prasad Raghavendra#### Hardness of Solving Sparse Overdetermined Linear Systems: A 3-Query PCP over Integers.

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TR09-001
| 26th November 2008
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Venkatesan Guruswami#### Artin automorphisms, Cyclotomic function fields, and Folded list-decodable codes

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TR08-105
| 26th November 2008
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Parikshit Gopalan, Venkatesan Guruswami, Prasad Raghavendra, Prasad Raghavendra#### List Decoding Tensor Products and Interleaved Codes

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TR08-054
| 13th May 2008
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Venkatesan Guruswami, Atri Rudra#### Concatenated codes can achieve list-decoding capacity

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TR08-036
| 14th March 2008
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Venkatesan Guruswami, Atri Rudra#### Soft decoding, dual BCH codes, and better list-decodable eps-biased codes

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TR08-008
| 8th February 2008
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Venkatesan Guruswami, Prasad Raghavendra#### Constraint Satisfaction over a Non-Boolean Domain: Approximation algorithms and Unique-Games hardness

Revisions: 1

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TR07-113
| 15th November 2007
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Matthew Andrews, Julia Chuzhoy, Venkatesan Guruswami, Sanjeev Khanna, Kunal Talwar, Lisa Zhang#### Inapproximability of edge-disjoint paths and low congestion routing on undirected graphs

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TR07-109
| 7th October 2007
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Venkatesan Guruswami, Atri Rudra#### Better Binary List-Decodable Codes via Multilevel Concatenation

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TR07-089
| 13th September 2007
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Parikshit Gopalan, Venkatesan Guruswami#### Deterministic Hardness Amplification via Local GMD Decoding

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TR07-086
| 7th September 2007
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Venkatesan Guruswami, James R. Lee, Alexander Razborov#### Almost Euclidean subspaces of $\ell_1^N$ via expander codes

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TR06-141
| 22nd November 2006
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Venkatesan Guruswami, Kunal Talwar#### Hardness of Low Congestion Routing in Directed Graphs

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TR06-134
| 18th October 2006
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Venkatesan Guruswami, Chris Umans, Salil Vadhan#### Extractors and condensers from univariate polynomials

Revisions: 1

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TR06-123
| 15th September 2006
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Venkatesan Guruswami, Venkatesan Guruswami#### Iterative Decoding of Low-Density Parity Check Codes (A Survey)

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TR06-123
| 15th September 2006
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Venkatesan Guruswami, Venkatesan Guruswami#### Iterative Decoding of Low-Density Parity Check Codes (A Survey)

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TR06-061
| 5th May 2006
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Venkatesan Guruswami, Prasad Raghavendra#### Hardness of Learning Halfspaces with Noise

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TR05-133
| 17th November 2005
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Venkatesan Guruswami, Atri Rudra#### Explicit Capacity-Achieving List-Decodable Codes

Revisions: 1

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TR05-132
| 8th November 2005
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Venkatesan Guruswami#### Algebraic-geometric generalizations of the Parvaresh-Vardy codes

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TR05-057
| 19th May 2005
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Venkatesan Guruswami, Valentine Kabanets#### Hardness amplification via space-efficient direct products

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TR05-019
| 9th February 2005
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Venkatesan Guruswami, Atri Rudra#### Tolerant Locally Testable Codes

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TR04-040
| 4th May 2004
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Venkatesan Guruswami, Alexander Vardy#### Maximum-likelihood decoding of Reed-Solomon codes is NP-hard

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TR03-080
| 12th November 2003
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Venkatesan Guruswami#### Better Extractors for Better Codes?

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TR02-053
| 20th July 2002
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Lars Engebretsen, Venkatesan Guruswami#### Is Constraint Satisfaction Over Two Variables Always Easy?

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TR02-027
| 30th April 2002
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Irit Dinur, Venkatesan Guruswami, Subhash Khot#### Vertex Cover on k-Uniform Hypergraphs is Hard to Approximate within Factor (k-3-\epsilon)

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TR01-002
| 6th December 2000
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Venkatesan Guruswami#### Constructions of Codes from Number Fields

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TR00-073
| 28th August 2000
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Venkatesan Guruswami, Sanjeev Khanna#### On the Hardness of 4-coloring a 3-colorable Graph

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TR00-062
| 25th August 2000
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Venkatesan Guruswami, Johan Håstad, Madhu Sudan#### Hardness of approximate hypergraph coloring

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TR99-043
| 5th November 1999
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Venkatesan Guruswami#### The Approximability of Set Splitting Problems and Satisfiability Problems with no Mixed Clauses

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TR98-043
| 27th July 1998
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Venkatesan Guruswami, Madhu Sudan#### Improved decoding of Reed-Solomon and algebraic-geometric codes.

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TR98-034
| 23rd June 1998
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Venkatesan Guruswami, Daniel Lewin and Madhu Sudan, Luca Trevisan#### A tight characterization of NP with 3 query PCPs

Omar Alrabiah, Venkatesan Guruswami, Ray Li

A simple, recently observed generalization of the classical Singleton bound to list-decoding asserts that rate $R$ codes are not list-decodable using list-size $L$ beyond an error fraction $\frac{L}{L+1} (1-R)$ (the Singleton bound being the case of $L=1$, i.e., unique decoding). We prove that in order to approach this bound for ... more >>>

Omar Alrabiah, Venkatesan Guruswami, Ray Li

Reed-Solomon codes are a classic family of error-correcting codes consisting of evaluations of low-degree polynomials over a finite field on some sequence of distinct field elements. They are widely known for their optimal unique-decoding capabilities, but their list-decoding capabilities are not fully understood. Given the prevalence of Reed-Solomon codes, a ... more >>>

Omar Alrabiah, Venkatesan Guruswami

A recent work of Li and Wootters (2021) shows a redundancy lower bound of $\Omega(\sqrt{Nk})$ for systematic linear $k$-batch codes of block length $N$ by looking at the $O(k)$ tensor power of the dual code. In this note, we present an alternate proof of their result via a linear independence ... more >>>

Venkatesan Guruswami, Jonathan Mosheiff

We prove the existence of Reed-Solomon codes of any desired rate $R \in (0,1)$ that are combinatorially list-decodable up to a radius approaching $1-R$, which is the information-theoretic limit. This is established by starting with the full-length $[q,k]_q$ Reed-Solomon code over a field $\mathbb{F}_q$ that is polynomially larger than the ... more >>>

Omar Alrabiah, Venkatesan Guruswami

We propose a framework to study the effect of local recovery requirements of codeword symbols on the dimension of linear codes, based on a combinatorial proxy that we call "visible rank." The locality constraints of a linear code are stipulated by a matrix $H$ of $\star$'s and $0$'s (which we ... more >>>

Vikraman Arvind, Venkatesan Guruswami

We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the Union-of-Subspace Avoidance (USA) problem), and finding a common zero of a system of ... more >>>

Venkatesan Guruswami, Xiaoyu He, Ray Li

We prove that there exists an absolute constant $\delta>0$ such any binary code $C\subset\{0,1\}^N$ tolerating $(1/2-\delta)N$ adversarial deletions must satisfy $|C|\le 2^{\poly\log N}$ and thus have rate asymptotically approaching $0$. This is the first constant fraction improvement over the trivial bound that codes tolerating $N/2$ adversarial deletions must have rate ... more >>>

Joshua Brakensiek, Venkatesan Guruswami, Sai Sandeep

Promise Constraint Satisfaction Problems (PCSPs) are a generalization of Constraint Satisfaction Problems (CSPs) where each predicate has a strong and a weak form and given a CSP instance, the objective is to distinguish if the strong form can be satisfied vs. even the weak form cannot be satisfied. Since their ... more >>>

Sivakanth Gopi, Venkatesan Guruswami

An $(n,r,h,a,q)$-Local Reconstruction Code is a linear code over $\mathbb{F}_q$ of length $n$, whose codeword symbols are partitioned into $n/r$ local groups each of size $r$. Each local group satisfies `$a$' local parity checks to recover from `$a$' erasures in that local group and there are further $h$ global parity ... more >>>

Venkatesan Guruswami, Chaoping Xing

We construct two classes of algebraic code families which are efficiently list decodable with small output list size from a fraction $1-R-\epsilon$ of adversarial errors where $R$ is the rate of the code, for any desired positive constant $\epsilon$. The alphabet size depends only on $\epsilon$ and is nearly-optimal.

The ... more >>>

Venkatesan Guruswami, Sai Sandeep

A famous conjecture of Tuza states that the minimum number of edges needed to cover all the triangles in a graph is at most twice the maximum number of edge-disjoint triangles. This conjecture was couched in a broader setting by Aharoni and Zerbib who proposed a hypergraph version of this ... more >>>

Venkatesan Guruswami, Vinayak Kumar

Random walks on expanders are a central and versatile tool in pseudorandomness. If an arbitrary half of the vertices of an expander graph are marked, known Chernoff bounds for expander walks imply that the number $M$ of marked vertices visited in a long $n$-step random walk strongly concentrates around the ... more >>>

Joshua Brakensiek, Venkatesan Guruswami, Marcin Wrochna, Stanislav Zivny

In the field of constraint satisfaction problems (CSP), promise CSPs are an exciting new direction of study. In a promise CSP, each constraint comes in two forms: "strict" and "weak," and in the associated decision problem one must distinguish between being able to satisfy all the strict constraints versus not ... more >>>

Venkatesan Guruswami, Andrii Riazanov, Min Ye

Let $W$ be a binary-input memoryless symmetric (BMS) channel with Shannon capacity $I(W)$ and fix any $\alpha > 0$. We construct, for any sufficiently small $\delta > 0$, binary linear codes of block length $O(1/\delta^{2+\alpha})$ and rate $I(W)-\delta$ that enable reliable communication on $W$ with quasi-linear time encoding and decoding. ... more >>>

Venkatesan Guruswami, Bernhard Haeupler, Amirbehshad Shahrasbi

We give a complete answer to the following basic question: ``What is the maximal fraction of deletions or insertions tolerable by $q$-ary list-decodable codes with non-vanishing information rate?''

This question has been open even for binary codes, including the restriction to the binary insertion-only setting, where the best known results ... more >>>

Venkatesan Guruswami, Sai Sandeep

The $d$-to-$1$ conjecture of Khot asserts that it is hard to satisfy an $\epsilon$ fraction of constraints of a satisfiable $d$-to-$1$ Label Cover instance, for arbitrarily small $\epsilon > 0$. We prove that the $d$-to-$1$ conjecture for any fixed $d$ implies the hardness of coloring a $4$-colorable graph with $C$ ... more >>>

Venkatesan Guruswami, Sai Sandeep

A $k$-uniform hypergraph is said to be $r$-rainbow colorable if there is an $r$-coloring of its vertices such that every hyperedge intersects all $r$ color classes. Given as input such a hypergraph, finding a $r$-rainbow coloring of it is NP-hard for all $k \ge 3$ and $r \ge 2$. ... more >>>

Venkatesan Guruswami, Jakub Opršal, Sai Sandeep

Dinur's celebrated proof of the PCP theorem alternates two main steps in several iterations: gap amplification to increase the soundness gap by a large constant factor (at the expense of much larger alphabet size), and a composition step that brings back the alphabet size to an absolute constant (at the ... more >>>

Joshua Brakensiek, Venkatesan Guruswami

Under the Strong Exponential Time Hypothesis, an integer linear program with $n$ Boolean-valued variables and $m$ equations cannot be solved in $c^n$ time for any constant $c < 2$. If the domain of the variables is relaxed to $[0,1]$, the associated linear program can of course be solved in polynomial ... more >>>

Joshua Brakensiek, Sivakanth Gopi, Venkatesan Guruswami

We study the complexity of Boolean constraint satisfaction problems (CSPs) when the assignment must have Hamming weight in some congruence class modulo $M$, for various choices of the modulus $M$. Due to the known classification of tractable Boolean CSPs, this mainly reduces to the study of three cases: 2SAT, HornSAT, ... more >>>

Omar Alrabiah, Venkatesan Guruswami

An $(n,k,\ell)$-vector MDS code is a $\mathbb{F}$-linear subspace of $(\mathbb{F}^\ell)^n$ (for some field $\mathbb{F}$) of dimension $k\ell$, such that any $k$ (vector) symbols of the codeword suffice to determine the remaining $r=n-k$ (vector) symbols. The length $\ell$ of each codeword symbol is called the sub-packetization of the code. Such a ... more >>>

Vijay Bhattiprolu, Mrinalkanti Ghosh, Venkatesan Guruswami, Euiwoong Lee, Madhur Tulsiani

We consider the $(\ell_p,\ell_r)$-Grothendieck problem, which seeks to maximize the bilinear form $y^T A x$ for an input matrix $A \in {\mathbb R}^{m \times n}$ over vectors $x,y$ with $\|x\|_p=\|y\|_r=1$. The problem is equivalent to computing the $p \to r^\ast$ operator norm of $A$, where $\ell_{r^*}$ is the dual norm ... more >>>

Venkatesan Guruswami, Andrii Riazanov

We say a subset $C \subseteq \{1,2,\dots,k\}^n$ is a $k$-hash code (also called $k$-separated) if for every subset of $k$ codewords from $C$, there exists a coordinate where all these codewords have distinct values. Understanding the largest possible rate (in bits), defined as $(\log_2 |C|)/n$, of a $k$-hash code is ... more >>>

Joshua Brakensiek, Venkatesan Guruswami

Promise CSPs are a relaxation of constraint satisfaction problems where the goal is to find an assignment satisfying a relaxed version of the constraints. Several well known problems can be cast as promise CSPs including approximate graph and hypergraph coloring, discrepancy minimization, and interesting variants of satisfiability. Similar to CSPs, ... more >>>

Vijay Bhattiprolu, Mrinalkanti Ghosh, Venkatesan Guruswami, Euiwoong Lee, Madhur Tulsiani

We study the problem of computing the $p\rightarrow q$ norm of a matrix $A \in R^{m \times n}$, defined as \[ \|A\|_{p\rightarrow q} ~:=~ \max_{x \,\in\, R^n \setminus \{0\}} \frac{\|Ax\|_q}{\|x\|_p} \] This problem generalizes the spectral norm of a matrix ($p=q=2$) and the Grothendieck problem ($p=\infty$, $q=1$), and has been ... more >>>

Jaroslaw Blasiok, Venkatesan Guruswami, Preetum Nakkiran, Atri Rudra, Madhu Sudan

Ar\i kan's exciting discovery of polar codes has provided an altogether new way to efficiently achieve Shannon capacity. Given a (constant-sized) invertible matrix $M$, a family of polar codes can be associated with this matrix and its ability to approach capacity follows from the $\textit{polarization}$ of an associated $[0,1]$-bounded martingale, ... more >>>

Venkatesan Guruswami, Nicolas Resch, Chaoping Xing

For a vector space $\mathbb{F}^n$ over a field $\mathbb{F}$, an $(\eta,\beta)$-dimension expander of degree $d$ is a collection of $d$ linear maps $\Gamma_j : \mathbb{F}^n \to \mathbb{F}^n$ such that for every subspace $U$ of $\mathbb{F}^n$ of dimension at most $\eta n$, the image of $U$ under all the maps, $\sum_{j=1}^d ... more >>>

Sivakanth Gopi, Venkatesan Guruswami, Sergey Yekhanin

In recent years the explosion in the volumes of data being stored online has resulted in distributed storage systems transitioning to erasure coding based schemes. Local Reconstruction Codes (LRCs) have emerged as the codes of choice for these applications. An $(n,r,h,a,q)$-LRC is a $q$-ary code, where encoding is as a ... more >>>

Venkatesan Guruswami, Rishi Saket

A hypergraph is $k$-rainbow colorable if there exists a vertex coloring using $k$ colors such that each hyperedge has all the $k$ colors. Unlike usual hypergraph coloring, rainbow coloring becomes harder as the number of colors increases. This work studies the rainbow colorability of hypergraphs which are guaranteed to be ... more >>>

Joshua Brakensiek, Venkatesan Guruswami

We give a family of dictatorship tests with perfect completeness and low-soundness for 2-to-2 constraints. The associated 2-to-2 conjecture has been the basis of some previous inapproximability results with perfect completeness. However, evidence towards the conjecture in the form of integrality gaps even against weak semidefinite programs has been elusive. ... more >>>

Joshua Brakensiek, Venkatesan Guruswami

The Unique Games Conjecture (UGC) has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect ... more >>>

Venkatesan Guruswami, Chaoping Xing, chen yuan

Subspace designs are a (large) collection of high-dimensional subspaces $\{H_i\}$ of $\F_q^m$ such that for any low-dimensional subspace $W$, only a small number of subspaces from the collection have non-trivial intersection with $W$; more precisely, the sum of dimensions of $W \cap H_i$ is at most some parameter $L$. The ... more >>>

Vijay Bhattiprolu, Mrinalkanti Ghosh, Venkatesan Guruswami, Euiwoong Lee, Madhur Tulsiani

We consider the following basic problem: given an $n$-variate degree-$d$ homogeneous polynomial $f$ with real coefficients, compute a unit vector $x \in \mathbb{R}^n$ that maximizes $|f(x)|$. Besides its fundamental nature, this problem arises in many diverse contexts ranging from tensor and operator norms to graph expansion to quantum information ... more >>>

Joshua Brakensiek, Venkatesan Guruswami

A classic result due to Schaefer (1978) classifies all constraint satisfaction problems (CSPs) over the Boolean domain as being either in $\mathsf{P}$ or NP-hard. This paper considers a promise-problem variant of CSPs called PCSPs. A PCSP over a finite set of pairs of constraints $\Gamma$ consists of a pair $(\Psi_P, ... more >>>

Venkatesan Guruswami, Jaikumar Radhakrishnan

Suppose Alice holds a uniformly random string $X \in \{0,1\}^N$ and Bob holds a noisy version $Y$ of $X$ where each bit of $X$ is flipped independently with probability $\epsilon \in [0,1/2]$. Alice and Bob would like to extract a common random string of min-entropy at least $k$. In this ... more >>>

Joshua Brakensiek, Venkatesan Guruswami

Finding a proper coloring of a $t$-colorable graph $G$ with $t$ colors is a classic NP-hard problem when $t\ge 3$. In this work, we investigate the approximate coloring problem in which the objective is to find a proper $c$-coloring of $G$ where $c \ge t$. We show that for all ... more >>>

Venkatesan Guruswami, Euiwoong Lee

The {\em Unique Coverage} problem, given a universe $V$ of elements and a collection $E$ of subsets of $V$, asks to find $S \subseteq V$ to maximize the number of $e \in E$ that intersects $S$ in {\em exactly one} element. When each $e \in E$ has cardinality at most ... more >>>

Boris Bukh, Venkatesan Guruswami

We consider codes over fixed alphabets against worst-case symbol deletions. For any fixed $k \ge 2$, we construct a family of codes over alphabet of size $k$ with positive rate, which allow efficient recovery from a worst-case deletion fraction approaching $1-\frac{2}{k+1}$. In particular, for binary codes, we are able to ... more >>>

Joshua Brakensiek, Venkatesan Guruswami, Samuel Zbarsky

We consider the problem of constructing binary codes to recover from $k$-bit deletions with efficient encoding/decoding, for a fixed $k$. The single deletion case is well understood, with the Varshamov-Tenengolts-Levenshtein code from 1965 giving an asymptotically optimal construction with $\approx 2^n/n$ codewords of length $n$, i.e., at most $\log n$ ... more >>>

Venkatesan Guruswami, Euiwoong Lee

A Boolean constraint satisfaction problem (CSP) is called approximation resistant if independently setting variables to $1$ with some probability $\alpha$ achieves the best possible approximation ratio for the fraction of constraints satisfied. We study approximation resistance of a natural subclass of CSPs that we call Symmetric Constraint Satisfaction Problems (SCSPs), ... more >>>

Venkatesan Guruswami, Ameya Velingker

We prove a lower estimate on the increase in entropy when two copies of a conditional random variable $X | Y$, with $X$ supported on $\mathbb{Z}_q=\{0,1,\dots,q-1\}$ for prime $q$, are summed modulo $q$. Specifically, given two i.i.d. copies $(X_1,Y_1)$ and $(X_2,Y_2)$ of a pair of random variables $(X,Y)$, with $X$ ... more >>>

Michael Forbes, Venkatesan Guruswami

An emerging theory of "linear-algebraic pseudorandomness" aims to understand the linear-algebraic analogs of fundamental Boolean pseudorandom objects where the rank of subspaces plays the role of the size of subsets. In this work, we study and highlight the interrelationships between several such algebraic objects such as subspace designs, dimension ... more >>>

Clement Canonne, Venkatesan Guruswami, Raghu Meka, Madhu Sudan

The communication complexity of many fundamental problems reduces greatly

when the communicating parties share randomness that is independent of the

inputs to the communication task. Natural communication processes (say between

humans) however often involve large amounts of shared correlations among the

communicating players, but rarely allow for perfect sharing of ...
more >>>

Venkatesan Guruswami, Madhu Sudan, Ameya Velingker, Carol Wang

Locally testable codes (LTCs) of constant distance that allow the tester to make a linear number of queries have become the focus of attention recently, due to their elegant connections to hardness of approximation. In particular, the binary Reed-Muller code of block length $N$ and distance $d$ is known to ... more >>>

Venkatesan Guruswami, Euiwoong Lee

Consider a $K$-uniform hypergraph $H = (V,E)$. A coloring $c : V \rightarrow \{1, 2, \dots, k \}$ with $k$ colors is rainbow if every hyperedge $e$ contains at least one vertex from each color, and is called perfectly balanced when each color appears the same number of times. A ... more >>>

Venkatesan Guruswami, Euiwoong Lee

The Feedback Vertex Set problem (FVS), where the goal is to find a small subset of vertices that intersects every cycle in an input directed graph, is among the fundamental problems whose approximability is not well-understood. One can efficiently find an $\widetilde{O}(\log n)$ factor approximation, and while a constant-factor approximation ... more >>>

Venkatesan Guruswami, Chaoping Xing

We give a length-efficient puncturing of Reed-Muller codes which preserves its distance properties. Formally, for the Reed-Muller code encoding $n$-variate degree-$d$ polynomials over ${\mathbb F}_q$ with $q \ge \Omega(d/\delta)$, we present an explicit (multi)-set $S \subseteq {\mathbb F}_q^n$ of size $N=\mathrm{poly}(n^d/\delta)$ such that every nonzero polynomial vanishes on at most ... more >>>

Venkatesan Guruswami, Carol Wang

We construct an explicit family of linear rank-metric codes over any field ${\mathbb F}_h$ that enables efficient list decoding up to a fraction $\rho$ of errors in the rank metric with a rate of $1-\rho-\epsilon$, for any desired $\rho \in (0,1)$ and $\epsilon > 0$. Previously, a Monte Carlo construction ... more >>>

Venkatesan Guruswami, Prahladh Harsha, Johan Håstad, Srikanth Srinivasan, Girish Varma

We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka, the “short code” of Barak et. al. [FOCS 2012]) and the techniques proposed by Dinur and Guruswami [FOCS 2013] to incorporate this code for inapproximability results.

In particular, we prove quasi-NP-hardness of the following problems on ... more >>>

Per Austrin, Venkatesan Guruswami, Johan Håstad

We prove the following hardness result for a natural promise variant of the classical CNF-satisfiability problem: Given a CNF-formula where each clause has width $w$ and the guarantee that there exists an assignment satisfying at least $g = \lceil \frac{w}{2}\rceil -1$ literals in each clause, it is NP-hard to find ... more >>>

Venkatesan Guruswami, Euiwoong Lee

We study two natural extensions of Constraint Satisfaction Problems (CSPs). {\em Balance}-Max-CSP requires that in any feasible assignment each element in the domain is used an equal number of times. An instance of {\em Hard}-Max-CSP consists of {\em soft constraints} and {\em hard constraints}, and the goal is to maximize ... more >>>

Irit Dinur, Venkatesan Guruswami

We develop new techniques to incorporate the recently proposed ``short code" (a low-degree version of the long code) into the construction and analysis of PCPs in the classical ``Label Cover + Fourier Analysis'' framework. As a result, we obtain more size-efficient PCPs that yield improved hardness results for approximating CSPs ... more >>>

Mahdi Cheraghchi, Venkatesan Guruswami

Non-malleable coding, introduced by Dziembowski, Pietrzak and Wichs (ICS 2010), aims for protecting the integrity of information against tampering attacks in situations where error-detection is impossible. Intuitively, information encoded by a non-malleable code either decodes to the original message or, in presence of any tampering, to an unrelated message. Non-malleable ... more >>>

Mahdi Cheraghchi, Venkatesan Guruswami

Non-malleable codes, introduced by Dziembowski, Pietrzak and Wichs (ICS 2010), encode messages $s$ in a manner so that tampering the codeword causes the decoder to either output $s$ or a message that is independent of $s$. While this is an impossible goal to achieve against unrestricted tampering functions, rather surprisingly ... more >>>

Venkatesan Guruswami, Sushant Sachdeva, Rishi Saket

We study the problem of computing the minimum vertex cover on $k$-uniform $k$-partite hypergraphs when the $k$-partition is given. On bipartite graphs ($k=2$), the minimum vertex cover can be computed in polynomial time. For $k \ge 3$, this problem is known to be NP-hard. For general $k$, the problem was ... more >>>

Venkatesan Guruswami, Swastik Kopparty

A subspace design is a collection $\{H_1,H_2,\dots,H_M\}$ of subspaces of ${\mathbf F}_q^m$ with the property that no low-dimensional subspace $W$ of ${\mathbf F}_q^m$ intersects too many subspaces of the collection. Subspace designs were introduced by Guruswami and Xing (STOC 2013) who used them to give a randomized construction of optimal ... more >>>

Venkatesan Guruswami, Patrick Xia

We prove that, for all binary-input symmetric memoryless channels, polar codes enable reliable communication at rates within $\epsilon > 0$ of the Shannon capacity with a block length, construction complexity, and decoding complexity all bounded by a *polynomial* in $1/\epsilon$. Polar coding gives the *first known explicit construction* with rigorous ... more >>>

Venkatesan Guruswami, Chaoping Xing

We construct a new list-decodable family of asymptotically good algebraic-geometric (AG) codes over fixed alphabets. The function fields underlying these codes are constructed using class field theory, specifically Drinfeld modules of rank $1$, and designed to have an automorphism of large order that is used to ``fold" the AG code. ... more >>>

Venkatesan Guruswami, Krzysztof Onak

We prove $n^{1+\Omega(1/p)}/p^{O(1)}$ lower bounds for the space complexity of $p$-pass streaming algorithms solving the following problems on $n$-vertex graphs:

* testing if an undirected graph has a perfect matching (this implies lower bounds for computing a maximum matching or even just the maximum matching size),

* testing if two ... more >>>

Venkatesan Guruswami, Chaoping Xing

We consider Reed-Solomon (RS) codes whose evaluation points belong to a subfield, and give a linear-algebraic list decoding algorithm that can correct a fraction of errors approaching the code distance, while pinning down the candidate messages to a well-structured affine space of dimension a constant factor smaller than the code ... more >>>

Venkatesan Guruswami, Ali Kemal Sinop

Convex relaxations based on different hierarchies of

linear/semi-definite programs have been used recently to devise

approximation algorithms for various optimization problems. The

approximation guarantee of these algorithms improves with the number

of {\em rounds} $r$ in the hierarchy, though the complexity of solving

(or even writing down the solution for) ...
more >>>

Mahdi Cheraghchi, Venkatesan Guruswami, Ameya Velingker

We prove that a random linear code over $\mathbb{F}_q$, with probability arbitrarily close to $1$, is list decodable at radius $1-1/q-\epsilon$ with list size $L=O(1/\epsilon^2)$ and rate $R=\Omega_q(\epsilon^2/(\log^3(1/\epsilon)))$. Up to the polylogarithmic factor in $1/\epsilon$ and constant factors depending on $q$, this matches the lower bound $L=\Omega_q(1/\epsilon^2)$ for the list ... more >>>

Venkatesan Guruswami, Yuan Zhou

A theorem of Håstad shows that for every constraint satisfaction problem (CSP) over a fixed size domain, instances where each variable appears in at most $O(1)$ constraints admit a non-trivial approximation algorithm, in the sense that one can beat (by an additive constant) the approximation ratio achieved by the naive ... more >>>

Venkatesan Guruswami, Carol Wang

Folded Reed-Solomon codes are an explicit family of codes that achieve the optimal trade-off between rate and list error-correction capability. Specifically, for any $\epsilon > 0$, Guruswami and Rudra presented an $n^{O(1/\epsilon)}$ time algorithm to list decode appropriate folded RS codes of rate $R$ from a fraction $1-R-\epsilon$ of ... more >>>

Venkatesan Guruswami, Chaoping Xing

We give a new construction of algebraic codes which are efficiently list decodable from a fraction $1-R-\epsilon$ of adversarial errors where $R$ is the rate of the code, for any desired positive constant $\epsilon$. The worst-case list size output by the algorithm is $O(1/\epsilon)$, matching the existential bound for random ... more >>>

Venkatesan Guruswami, Srivatsan Narayanan

We prove the following results concerning the combinatorics of list decoding, motivated by the exponential gap between the known upper bound (of $O(1/\gamma)$) and lower bound (of $\Omega_p(\log (1/\gamma))$) for the list-size needed to decode up to radius $p$ with rate $\gamma$ away from capacity, i.e., $1-h(p)-\gamma$ (here $p\in (0,1/2)$ ... more >>>

Venkatesan Guruswami, Ali Kemal Sinop

We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and Small Set expansion, as well as the Unique Games problem. These ... more >>>

Venkatesan Guruswami, Johan Håstad, Rajsekar Manokaran, Prasad Raghavendra, Moses Charikar

We prove that, assuming the Unique Games Conjecture (UGC), every problem in the class of ordering constraint satisfaction problems (OCSP) where each constraint has constant arity is approximation

resistant. In other words, we show that if $\rho$ is the expected fraction of constraints satisfied by a random ordering, then obtaining ...
more >>>

Vitaly Feldman, Venkatesan Guruswami, Prasad Raghavendra, Yi Wu

We prove the following strong hardness result for learning: Given a distribution of labeled examples from the hypercube such that there exists a monomial consistent with $(1-\epsilon)$ of the examples, it is $\mathrm{NP}$-hard to find a halfspace that is correct on $(1/2+\epsilon)$ of the examples, for arbitrary constants $\epsilon ... more >>>

Venkatesan Guruswami, Prasad Raghavendra, Rishi Saket, Yi Wu

The Unique Games conjecture (UGC) has emerged in recent years as the starting point for several optimal inapproximability results. While for none of these results a reverse reduction to Unique Games is known, the assumption of bijective projections in the Label Cover instance seems critical in these proofs. In this ... more >>>

Venkatesan Guruswami, Ali Kemal Sinop

We prove almost tight hardness results for finding independent sets in bounded degree graphs and hypergraphs that admit a good

coloring. Our specific results include the following (where $\Delta$, assumed to be a constant, is a bound on the degree, and

$n$ is the number of vertices):

Venkatesan Guruswami, Adam Smith

In this paper, we consider coding schemes for computationally bounded channels, which can introduce an arbitrary set of errors as long as (a) the fraction of errors is bounded with high probability by a parameter p and (b) the process which adds the errors can be described by a sufficiently ... more >>>

Venkatesan Guruswami, Yuan Zhou

We study the approximability of two natural Boolean constraint satisfaction problems: Horn satisfiability and exact hitting set. Under the Unique Games conjecture, we prove the following optimal inapproximability and approximability results for finding an assignment satisfying as many constraints as possible given a {\em

near-satisfiable} instance.

\begin{enumerate}

\item ...
more >>>

Venkatesan Guruswami, Johan Håstad, Swastik Kopparty

For every fixed finite field $\F_q$, $p \in (0,1-1/q)$ and $\varepsilon >

0$, we prove that with high probability a random subspace $C$ of

$\F_q^n$ of dimension $(1-H_q(p)-\varepsilon)n$ has the

property that every Hamming ball of radius $pn$ has at most

$O(1/\varepsilon)$ codewords.

This ... more >>>

Eli Ben-Sasson, Venkatesan Guruswami, Tali Kaufman, Madhu Sudan, Michael Viderman

Locally testable codes (LTCs) are error-correcting codes for which membership, in the code, of a given word can be tested by examining it in very few locations. Most known constructions of locally testable codes are linear codes, and give error-correcting codes

whose duals have (superlinearly) {\em many} small weight ...
more >>>

Venkatesan Guruswami, Ali Kemal Sinop

We study the maximization version of the fundamental graph coloring problem. Here the goal is to color the vertices of a $k$-colorable graph with $k$ colors so that a maximum fraction of edges are properly colored (i.e., their endpoints receive different colors). A random $k$-coloring properly colors an expected fraction ... more >>>

Venkatesan Guruswami, Prasad Raghavendra

A classic result due to Hastad established that for every constant \eps > 0, given an overdetermined system of linear equations over a finite field \F_q where each equation depends on exactly 3 variables and at least a fraction (1-\eps) of the equations can be satisfied, it is NP-hard to ... more >>>

Venkatesan Guruswami

Algebraic codes that achieve list decoding capacity were recently

constructed by a careful ``folding'' of the Reed-Solomon code. The

``low-degree'' nature of this folding operation was crucial to the list

decoding algorithm. We show how such folding schemes conducive to list

decoding arise out of the Artin-Frobenius automorphism at primes ...
more >>>

Parikshit Gopalan, Venkatesan Guruswami, Prasad Raghavendra, Prasad Raghavendra

We design the first efficient algorithms and prove new combinatorial bounds for list decoding tensor products of codes and interleaved codes.

1)We show that for every code, the ratio of its list decoding radius to its minimum distance stays unchanged under the tensor product operation (rather than squaring, as one ... more >>>

Venkatesan Guruswami, Atri Rudra

We prove that binary linear concatenated codes with an outer algebraic code (specifically, a folded Reed-Solomon code) and independently and randomly chosen linear inner codes achieve the list-decoding capacity with high probability. In particular, for any $0 < \rho < 1/2$ and $\epsilon > 0$, there exist concatenated codes of ... more >>>

Venkatesan Guruswami, Atri Rudra

We construct binary linear codes that are efficiently list-decodable

up to a fraction $(1/2-\eps)$ of errors. The codes encode $k$ bits

into $n = {\rm poly}(k/\eps)$ bits and are constructible and

list-decodable in time polynomial in $k$ and $1/\eps$ (in

particular, in our results $\eps$ need ...
more >>>

Venkatesan Guruswami, Prasad Raghavendra

We study the approximability of the \maxcsp problem over non-boolean domains, more specifically over $\{0,1,\ldots,q-1\}$ for some integer $q$. We obtain a approximation algorithm that achieves a ratio of $C(q) \cdot k/q^k$ for some constant $C(q)$ depending only on $q$. Further, we extend the techniques of Samorodnitsky and Trevisan to ... more >>>

Matthew Andrews, Julia Chuzhoy, Venkatesan Guruswami, Sanjeev Khanna, Kunal Talwar, Lisa Zhang

In the undirected Edge-Disjoint Paths problem with Congestion

(EDPwC), we are given an undirected graph with $V$ nodes, a set of

terminal pairs and an integer $c$. The objective is to route as many

terminal pairs as possible, subject to the constraint that at most

$c$ demands can be routed ...
more >>>

Venkatesan Guruswami, Atri Rudra

We give a polynomial time construction of binary codes with the best

currently known trade-off between rate and error-correction

radius. Specifically, we obtain linear codes over fixed alphabets

that can be list decoded in polynomial time up to the so called

Blokh-Zyablov bound. Our work ...
more >>>

Parikshit Gopalan, Venkatesan Guruswami

We study the average-case hardness of the class NP against

deterministic polynomial time algorithms. We prove that there exists

some constant $\mu > 0$ such that if there is some language in NP

for which no deterministic polynomial time algorithm can decide L

correctly on a $1- (log n)^{-\mu}$ fraction ...
more >>>

Venkatesan Guruswami, James R. Lee, Alexander Razborov

We give an explicit (in particular, deterministic polynomial time)

construction of subspaces $X

\subseteq \R^N$ of dimension $(1-o(1))N$ such that for every $x \in X$,

$$(\log N)^{-O(\log\log\log N)} \sqrt{N}\, \|x\|_2 \leq \|x\|_1 \leq \sqrt{N}\, \|x\|_2.$$

If we are allowed to use $N^{1/\log\log N}\leq N^{o(1)}$ random bits

and ...
more >>>

Venkatesan Guruswami, Kunal Talwar

We prove a strong inapproximability result for routing on directed

graphs with low congestion. Given as input a directed graph on $N$

vertices and a set of source-destination pairs that can be connected

via edge-disjoint paths, we prove that it is hard, assuming NP

doesn't have $n^{O(\log\log n)}$ time randomized ...
more >>>

Venkatesan Guruswami, Chris Umans, Salil Vadhan

We give new constructions of randomness extractors and lossless condensers that are optimal to within constant factors in both the seed length and the output length. For extractors, this matches the parameters of the current best known construction [LRVW03]; for lossless condensers, the previous best constructions achieved optimality to within ... more >>>

Venkatesan Guruswami, Venkatesan Guruswami

Much progress has been made on decoding algorithms for

error-correcting codes in the last decade. In this article, we give an

introduction to some fundamental results on iterative, message-passing

algorithms for low-density parity check codes. For certain

important stochastic channels, this line of work has enabled getting

very close to ...
more >>>

Venkatesan Guruswami, Venkatesan Guruswami

error-correcting codes in the last decade. In this article, we give an

introduction to some fundamental results on iterative, message-passing

algorithms for low-density parity check codes. For certain

important stochastic channels, this line of work has enabled getting

very close to ...
more >>>

Venkatesan Guruswami, Prasad Raghavendra

Learning an unknown halfspace (also called a perceptron) from

labeled examples is one of the classic problems in machine learning.

In the noise-free case, when a halfspace consistent with all the

training examples exists, the problem can be solved in polynomial

time using linear programming. ...
more >>>

Venkatesan Guruswami, Atri Rudra

For every $0 < R < 1$ and $\eps > 0$, we present an explicit

construction of error-correcting codes of rate $R$ that can be list

decoded in polynomial time up to a fraction $(1-R-\eps)$ of errors.

These codes achieve the ``capacity'' for decoding from {\em ...
more >>>

Venkatesan Guruswami

This paper is concerned with a new family of error-correcting codes

based on algebraic curves over finite fields, and list decoding

algorithms for them. The basic goal in the subject of list decoding is

to construct error-correcting codes $C$ over some alphabet $\Sigma$

which have good rate $R$, and at ...
more >>>

Venkatesan Guruswami, Valentine Kabanets

We prove a version of the derandomized Direct Product Lemma for

deterministic space-bounded algorithms. Suppose a Boolean function

$g:\{0,1\}^n\to\{0,1\}$ cannot be computed on more than $1-\delta$

fraction of inputs by any deterministic time $T$ and space $S$

algorithm, where $\delta\leq 1/t$ for some $t$. Then, for $t$-step

walks $w=(v_1,\dots, v_t)$ ...
more >>>

Venkatesan Guruswami, Atri Rudra

An error-correcting code is said to be {\em locally testable} if it has an

efficient spot-checking procedure that can distinguish codewords

from strings that are far from every codeword, looking at very few

locations of the input in doing so. Locally testable codes (LTCs) have

generated ...
more >>>

Venkatesan Guruswami, Alexander Vardy

Maximum-likelihood decoding is one of the central algorithmic

problems in coding theory. It has been known for over 25 years

that maximum-likelihood decoding of general linear codes is

NP-hard. Nevertheless, it was so far unknown whether maximum-

likelihood decoding remains hard for any specific family of

more >>>

Venkatesan Guruswami

We present an explicit construction of codes that can be list decoded

from a fraction $(1-\eps)$ of errors in sub-exponential time and which

have rate $\eps/\log^{O(1)}(1/\eps)$. This comes close to the optimal

rate of $\Omega(\eps)$, and is the first sub-exponential complexity

construction to beat the rate of $O(\eps^2)$ achieved by ...
more >>>

Lars Engebretsen, Venkatesan Guruswami

By the breakthrough work of Håstad, several constraint satisfaction

problems are now known to have the following approximation resistance

property: satisfying more clauses than what picking a random

assignment would achieve is NP-hard. This is the case for example for

Max E3-Sat, Max E3-Lin and Max E4-Set Splitting. A notable ...
more >>>

Irit Dinur, Venkatesan Guruswami, Subhash Khot

Given a $k$-uniform hypergraph, the E$k$-Vertex-Cover problem is

to find a minimum subset of vertices that ``hits'' every edge. We

show that for every integer $k \geq 5$, E$k$-Vertex-Cover is

NP-hard to approximate within a factor of $(k-3-\epsilon)$, for

an arbitrarily small constant $\epsilon > 0$.

This almost matches the ... more >>>

Venkatesan Guruswami

We define number-theoretic error-correcting codes based on algebraic

number fields, thereby providing a generalization of Chinese Remainder

Codes akin to the generalization of Reed-Solomon codes to

Algebraic-geometric codes. Our construction is very similar to

(and in fact less general than) the one given by (Lenstra 1986), but

the ...
more >>>

Venkatesan Guruswami, Sanjeev Khanna

We give a new proof showing that it is NP-hard to color a 3-colorable

graph using just four colors. This result is already known (Khanna,

Linial, Safra 1992), but our proof is novel as it does not rely on

the PCP theorem, while the earlier one does. This ...
more >>>

Venkatesan Guruswami, Johan Håstad, Madhu Sudan

We introduce the notion of covering complexity of a probabilistic

verifier. The covering complexity of a verifier on a given input is

the minimum number of proofs needed to ``satisfy'' the verifier on

every random string, i.e., on every random string, at least one of the

given proofs must be ...
more >>>

Venkatesan Guruswami

We prove hardness results for approximating set splitting problems and

also instances of satisfiability problems which have no ``mixed'' clauses,

i.e all clauses have either all their literals unnegated or all of them

negated. Recent results of Hastad imply tight hardness results for set

splitting when all sets ...
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Venkatesan Guruswami, Madhu Sudan

We present an improved list decoding algorithm for decoding

Reed-Solomon codes. Given an arbitrary string of length n, the

list decoding problem is that of finding all codewords within a

specified Hamming distance from the input string.

It is well-known that this decoding problem for Reed-Solomon

codes reduces to the ...
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Venkatesan Guruswami, Daniel Lewin and Madhu Sudan, Luca Trevisan

It is known that there exists a PCP characterization of NP

where the verifier makes 3 queries and has a {\em one-sided}

error that is bounded away from 1; and also that 2 queries

do not suffice for such a characterization. Thus PCPs with

3 ...
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