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**Published in:** LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Rossman [In Proc. 34th Comput. Complexity Conf., 2019] introduced the notion of criticality. The criticality of a Boolean function f : {0,1}ⁿ → {0,1} is the minimum λ ≥ 1 such that for all positive integers t and all p ∈ [0,1], Pr_{ρ∼ℛ_p}[DT_{depth}(f|_ρ) ≥ t] ≤ (pλ)^t, where ℛ_p refers to the distribution of p-random restrictions.
Håstad’s celebrated switching lemma shows that the criticality of any k-DNF is at most O(k). Subsequent improvements to correlation bounds of AC⁰-circuits against parity showed that the criticality of any AC⁰-circuit of size S and depth d+1 is at most O(log S)^d and any regular AC⁰-formula of size S and depth d+1 is at most O((1/d)⋅log S)^d. We strengthen these results by showing that the criticality of any AC⁰-formula (not necessarily regular) of size S and depth d+1 is at most O((log S)/d)^d, resolving a conjecture due to Rossman.
This result also implies Rossman’s optimal lower bound on the size of any depth-d AC⁰-formula computing parity [Comput. Complexity, 27(2):209-223, 2018.]. Our result implies tight correlation bounds against parity, tight Fourier concentration results and improved #SAT algorithm for AC⁰-formulae.

Prahladh Harsha, Tulasimohan Molli, and Ashutosh Shankar. Criticality of AC⁰-Formulae. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 19:1-19:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{harsha_et_al:LIPIcs.CCC.2023.19, author = {Harsha, Prahladh and Molli, Tulasimohan and Shankar, Ashutosh}, title = {{Criticality of AC⁰-Formulae}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {19:1--19:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-282-2}, ISSN = {1868-8969}, year = {2023}, volume = {264}, editor = {Ta-Shma, Amnon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.19}, URN = {urn:nbn:de:0030-drops-182898}, doi = {10.4230/LIPIcs.CCC.2023.19}, annote = {Keywords: AC⁰ circuits, AC⁰ formulae, criticality, switching lemma, correlation bounds} }

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

A problem is downward self-reducible if it can be solved efficiently given an oracle that returns solutions for strictly smaller instances. In the decisional landscape, downward self-reducibility is well studied and it is known that all downward self-reducible problems are in PSPACE. In this paper, we initiate the study of downward self-reducible search problems which are guaranteed to have a solution - that is, the downward self-reducible problems in TFNP. We show that most natural PLS-complete problems are downward self-reducible and any downward self-reducible problem in TFNP is contained in PLS. Furthermore, if the downward self-reducible problem is in TFUP (i.e. it has a unique solution), then it is actually contained in UEOPL, a subclass of CLS. This implies that if integer factoring is downward self-reducible then it is in fact in UEOPL, suggesting that no efficient factoring algorithm exists using the factorization of smaller numbers.

Prahladh Harsha, Daniel Mitropolsky, and Alon Rosen. Downward Self-Reducibility in TFNP. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 67:1-67:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{harsha_et_al:LIPIcs.ITCS.2023.67, author = {Harsha, Prahladh and Mitropolsky, Daniel and Rosen, Alon}, title = {{Downward Self-Reducibility in TFNP}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {67:1--67:17}, 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.2023.67}, URN = {urn:nbn:de:0030-drops-175700}, doi = {10.4230/LIPIcs.ITCS.2023.67}, annote = {Keywords: downward self-reducibility, TFNP, TFUP, factoring, PLS, CLS} }

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

We study the following natural question on random sets of points in 𝔽₂^m:
Given a random set of k points Z = {z₁, z₂, … , z_k} ⊆ 𝔽₂^m, what is the dimension of the space of degree at most r multilinear polynomials that vanish on all points in Z?
We show that, for r ≤ γ m (where γ > 0 is a small, absolute constant) and k = (1-ε)⋅binom(m, ≤ r) for any constant ε > 0, the space of degree at most r multilinear polynomials vanishing on a random set Z = {z_1,…, z_k} has dimension exactly binom(m, ≤ r) - k with probability 1 - o(1). This bound shows that random sets have a much smaller space of degree at most r multilinear polynomials vanishing on them, compared to the worst-case bound (due to Wei (IEEE Trans. Inform. Theory, 1991)) of binom(m, ≤ r) - binom(log₂ k, ≤ r) ≫ binom(m, ≤ r) - k.
Using this bound, we show that high-degree Reed-Muller codes (RM(m,d) with d > (1-γ) m) "achieve capacity" under the Binary Erasure Channel in the sense that, for any ε > 0, we can recover from (1-ε)⋅binom(m, ≤ m-d-1) random erasures with probability 1 - o(1). This also implies that RM(m,d) is also efficiently decodable from ≈ binom(m, ≤ m-(d/2)) random errors for the same range of parameters.

Siddharth Bhandari, Prahladh Harsha, Ramprasad Saptharishi, and Srikanth Srinivasan. Vanishing Spaces of Random Sets and Applications to Reed-Muller Codes. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bhandari_et_al:LIPIcs.CCC.2022.31, author = {Bhandari, Siddharth and Harsha, Prahladh and Saptharishi, Ramprasad and Srinivasan, Srikanth}, title = {{Vanishing Spaces of Random Sets and Applications to Reed-Muller Codes}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {31:1--31:14}, 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.2022.31}, URN = {urn:nbn:de:0030-drops-165934}, doi = {10.4230/LIPIcs.CCC.2022.31}, annote = {Keywords: Reed-Muller codes, polynomials, weight-distribution, vanishing ideals, erasures, capacity} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

In this paper, we show the mixing of three-term progressions (x, xg, xg²) in every finite quasirandom group, fully answering a question of Gowers. More precisely, we show that for any D-quasirandom group G and any three sets A₁, A₂, A₃ ⊂ G, we have
|Pr_{x,y∼ G}[x ∈ A₁, xy ∈ A₂, xy² ∈ A₃] - ∏_{i = 1}³ Pr_{x∼ G}[x ∈ A_i]| ≤ (2/(√{D)})^{1/4}.
Prior to this, Tao answered this question when the underlying quasirandom group is SL_{d}(𝔽_q). Subsequently, Peluse extended the result to all non-abelian finite simple groups. In this work, we show that a slight modification of Peluse’s argument is sufficient to fully resolve Gowers' quasirandom conjecture for 3-term progressions. Surprisingly, unlike the proofs of Tao and Peluse, our proof is elementary and only uses basic facts from non-abelian Fourier analysis.

Amey Bhangale, Prahladh Harsha, and Sourya Roy. Mixing of 3-Term Progressions in Quasirandom Groups. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 20:1-20:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bhangale_et_al:LIPIcs.ITCS.2022.20, author = {Bhangale, Amey and Harsha, Prahladh and Roy, Sourya}, title = {{Mixing of 3-Term Progressions in Quasirandom Groups}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {20:1--20:9}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.20}, URN = {urn:nbn:de:0030-drops-156163}, doi = {10.4230/LIPIcs.ITCS.2022.20}, annote = {Keywords: Quasirandom groups, 3-term arithmetic progressions} }

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RANDOM

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

In this work, we present an abstract framework for some algebraic error-correcting codes with the aim of capturing codes that are list-decodable to capacity, along with their decoding algorithm. In the polynomial ideal framework, a code is specified by some ideals in a polynomial ring, messages are polynomials and their encoding is the residue modulo the ideals. We present an alternate way of viewing this class of codes in terms of linear operators, and show that this alternate view makes their algorithmic list-decodability amenable to analysis.
Our framework leads to a new class of codes that we call affine Folded Reed-Solomon codes (which are themselves a special case of the broader class we explore). These codes are common generalizations of the well-studied Folded Reed-Solomon codes and Univariate Multiplicity codes, while also capturing the less-studied Additive Folded Reed-Solomon codes as well as a large family of codes that were not previously known/studied.
More significantly our framework also captures the algorithmic list-decodability of the constituent codes. Specifically, we present a unified view of the decoding algorithm for ideal-theoretic codes and show that the decodability reduces to the analysis of the distance of some related codes. We show that good bounds on this distance lead to capacity-achieving performance of the underlying code, providing a unifying explanation of known capacity-achieving results. In the specific case of affine Folded Reed-Solomon codes, our framework shows that they are list-decodable up to capacity (for appropriate setting of the parameters), thereby unifying the previous results for Folded Reed-Solomon, Multiplicity and Additive Folded Reed-Solomon codes.

Siddharth Bhandari, Prahladh Harsha, Mrinal Kumar, and Madhu Sudan. Ideal-Theoretic Explanation of Capacity-Achieving Decoding. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 56:1-56:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bhandari_et_al:LIPIcs.APPROX/RANDOM.2021.56, author = {Bhandari, Siddharth and Harsha, Prahladh and Kumar, Mrinal and Sudan, Madhu}, title = {{Ideal-Theoretic Explanation of Capacity-Achieving Decoding}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {56:1--56:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.56}, URN = {urn:nbn:de:0030-drops-147499}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.56}, annote = {Keywords: List Decodability, List Decoding Capacity, Polynomial Ideal Codes, Multiplicity Codes, Folded Reed-Solomon Codes} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

We construct an explicit and structured family of 3XOR instances which is hard for O(√{log n}) levels of the Sum-of-Squares hierarchy. In contrast to earlier constructions, which involve a random component, our systems are highly structured and can be constructed explicitly in deterministic polynomial time.
Our construction is based on the high-dimensional expanders devised by Lubotzky, Samuels and Vishne, known as LSV complexes or Ramanujan complexes, and our analysis is based on two notions of expansion for these complexes: cosystolic expansion, and a local isoperimetric inequality due to Gromov.
Our construction offers an interesting contrast to the recent work of Alev, Jeronimo and the last author (FOCS 2019). They showed that 3XOR instances in which the variables correspond to vertices in a high-dimensional expander are easy to solve. In contrast, in our instances the variables correspond to the edges of the complex.

Irit Dinur, Yuval Filmus, Prahladh Harsha, and Madhur Tulsiani. Explicit SoS Lower Bounds from High-Dimensional Expanders. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dinur_et_al:LIPIcs.ITCS.2021.38, author = {Dinur, Irit and Filmus, Yuval and Harsha, Prahladh and Tulsiani, Madhur}, title = {{Explicit SoS Lower Bounds from High-Dimensional Expanders}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {38:1--38:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.38}, URN = {urn:nbn:de:0030-drops-135774}, doi = {10.4230/LIPIcs.ITCS.2021.38}, annote = {Keywords: High-dimensional expanders, sum-of-squares, integrality gaps} }

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RANDOM

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

In this work, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over F₂. We show the following results for multilinear forms and tensors.
Correlation bounds. We show that a random d-linear form has exponentially low correlation with low-degree polynomials. More precisely, for d = 2^{o(k)}, we show that a random d-linear form f(X₁,X₂, … , X_d) : (F₂^{k}) ^d → F₂ has correlation 2^{-k(1-o(1))} with any polynomial of degree at most d/2 with high probability. This result is proved by giving near-optimal bounds on the bias of a random d-linear form, which is in turn proved by giving near-optimal bounds on the probability that a sum of t random d-dimensional rank-1 tensors is identically zero.
Tensor rank vs Bias. We show that if a 3-dimensional tensor has small rank then its bias, when viewed as a 3-linear form, is large. More precisely, given any 3-dimensional tensor T: [k]³ → F₂ of rank at most t, the bias of the 3-linear form f_T(X₁, X₂, X₃) : = ∑_{(i₁, i₂, i₃) ∈ [k]³} T(i₁, i₂, i₃)⋅ X_{1,i₁}⋅ X_{2,i₂}⋅ X_{3,i₃} is at least (3/4)^t. This bias vs tensor-rank connection suggests a natural approach to proving nontrivial tensor-rank lower bounds. In particular, we use this approach to give a new proof that the finite field multiplication tensor has tensor rank at least 3.52 k, which is the best known rank lower bound for any explicit tensor in three dimensions over F₂. Moreover, this relation between bias and tensor rank holds for d-dimensional tensors for any fixed d.

Abhishek Bhrushundi, Prahladh Harsha, Pooya Hatami, Swastik Kopparty, and Mrinal Kumar. On Multilinear Forms: Bias, Correlation, and Tensor Rank. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 29:1-29:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bhrushundi_et_al:LIPIcs.APPROX/RANDOM.2020.29, author = {Bhrushundi, Abhishek and Harsha, Prahladh and Hatami, Pooya and Kopparty, Swastik and Kumar, Mrinal}, title = {{On Multilinear Forms: Bias, Correlation, and Tensor Rank}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {29:1--29:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.29}, URN = {urn:nbn:de:0030-drops-126325}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.29}, annote = {Keywords: polynomials, Boolean functions, tensor rank, bias, correlation} }

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APPROX

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

We prove that for every constant c and epsilon = (log n)^{-c}, there is no polynomial time algorithm that when given an instance of 3-LIN with n variables where an (1 - epsilon)-fraction of the clauses are satisfiable, finds an assignment that satisfies atleast (1/2 + epsilon)-fraction of clauses unless NP subseteq BPP. The previous best hardness using a polynomial time reduction achieves epsilon = (log log n)^{-c}, which is obtained by the Label Cover hardness of Moshkovitz and Raz [J. ACM, 57(5), 2010] followed by the reduction from Label Cover to 3-LIN of Håstad [J. ACM, 48(4):798 - 859, 2001].
Our main idea is to prove a hardness result for Label Cover similar to Moshkovitz and Raz where each projection has a linear structure. This linear structure of Label Cover allows us to use Hadamard codes instead of long codes, making the reduction more efficient. For the hardness of Linear Label Cover, we follow the work of Dinur and Harsha [SIAM J. Comput., 42(6):2452 - 2486, 2013] that simplified the construction of Moshkovitz and Raz, and observe that running their reduction from a hardness of the problem LIN (of unbounded arity) instead of the more standard problem of solving quadratic equations ensures the linearity of the resultant Label Cover.

Prahladh Harsha, Subhash Khot, Euiwoong Lee, and Devanathan Thiruvenkatachari. Improved 3LIN Hardness via Linear Label Cover. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{harsha_et_al:LIPIcs.APPROX-RANDOM.2019.9, author = {Harsha, Prahladh and Khot, Subhash and Lee, Euiwoong and Thiruvenkatachari, Devanathan}, title = {{Improved 3LIN Hardness via Linear Label Cover}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {9:1--9:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.9}, URN = {urn:nbn:de:0030-drops-112245}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.9}, annote = {Keywords: probabilistically checkable proofs, PCP, composition, 3LIN, low soundness error} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

A local tester for an error-correcting code is a probabilistic procedure that queries a small subset of coordinates, accepts codewords with probability one, and rejects non-codewords with probability proportional to their distance from the code. The local tester is robust if for non-codewords it satisfies the stronger property that the average distance of local views from accepting views is proportional to the distance from the code. Robust testing is an important component in constructions of locally testable codes and probabilistically checkable proofs as it allows for composition of local tests.
In this work we show that for certain codes, any (natural) local tester can be converted to a roubst tester with roughly the same number of queries. Our result holds for the class of affine-invariant lifted codes which is a broad class of codes that includes Reed-Muller codes, as well as recent constructions of high-rate locally testable codes (Guo, Kopparty, and Sudan, ITCS 2013). Instantiating this with known local testing results for lifted codes gives a more direct proof that improves some of the parameters of the main result of Guo, Haramaty, and Sudan (FOCS 2015), showing robustness of lifted codes.
To obtain the above transformation we relate the notions of local testing and robust testing to the notion of agreement testing that attempts to find out whether valid partial assignments can be stitched together to a global codeword. We first show that agreement testing implies robust testing, and then show that local testing implies agreement testing. Our proof is combinatorial, and is based on expansion / sampling properties of the collection of local views of local testers. Thus, it immediately applies to local testers of lifted codes that query random affine subspaces in F_q^m, and moreover seems amenable to extension to other families of locally testable codes with expanding families of local views.

Irit Dinur, Prahladh Harsha, Tali Kaufman, and Noga Ron-Zewi. From Local to Robust Testing via Agreement Testing. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dinur_et_al:LIPIcs.ITCS.2019.29, author = {Dinur, Irit and Harsha, Prahladh and Kaufman, Tali and Ron-Zewi, Noga}, title = {{From Local to Robust Testing via Agreement Testing}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {29:1--29:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.29}, URN = {urn:nbn:de:0030-drops-101221}, doi = {10.4230/LIPIcs.ITCS.2019.29}, annote = {Keywords: Local testing, Robust testing, Agreement testing, Affine-invariant codes, Lifted codes} }

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**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

We study the probabilistic degree over R of the OR function on n variables. For epsilon in (0,1/3), the epsilon-error probabilistic degree of any Boolean function f:{0,1}^n -> {0,1} over R is the smallest non-negative integer d such that the following holds: there exists a distribution of polynomials Pol in R[x_1,...,x_n] entirely supported on polynomials of degree at most d such that for all z in {0,1}^n, we have Pr_{P ~ Pol}[P(z) = f(z)] >= 1- epsilon. It is known from the works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. 6th CCC 1991), that the epsilon-error probabilistic degree of the OR function is at most O(log n * log(1/epsilon)). Our first observation is that this can be improved to O{log (n atop <= log(1/epsilon))}, which is better for small values of epsilon.
In all known constructions of probabilistic polynomials for the OR function (including the above improvement), the polynomials P in the support of the distribution Pol have the following special structure: P(x_1,...,x_n) = 1 - prod_{i in [t]} (1- L_i(x_1,...,x_n)), where each L_i(x_1,..., x_n) is a linear form in the variables x_1,...,x_n, i.e., the polynomial 1-P(bar{x}) is a product of affine forms. We show that the epsilon-error probabilistic degree of OR when restricted to polynomials of the above form is Omega(log (n over <= log(1/epsilon))/log^2 (log (n over <= log(1/epsilon))})), thus matching the above upper bound (up to polylogarithmic factors).

Siddharth Bhandari, Prahladh Harsha, Tulasimohan Molli, and Srikanth Srinivasan. On the Probabilistic Degree of OR over the Reals. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 5:1-5:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bhandari_et_al:LIPIcs.FSTTCS.2018.5, author = {Bhandari, Siddharth and Harsha, Prahladh and Molli, Tulasimohan and Srinivasan, Srikanth}, title = {{On the Probabilistic Degree of OR over the Reals}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {5:1--5:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.5}, URN = {urn:nbn:de:0030-drops-99044}, doi = {10.4230/LIPIcs.FSTTCS.2018.5}, annote = {Keywords: Polynomials over reals, probabilistic polynomials, probabilistic degree, OR polynomial} }

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**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

We initiate the study of Boolean function analysis on high-dimensional expanders. We describe an analog of the Fourier expansion and of the Fourier levels on simplicial complexes, and generalize the FKN theorem to high-dimensional expanders.
Our results demonstrate that a high-dimensional expanding complex X can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing |X(k)|=O(n) points in comparison to binom{n}{k+1} points in the (k+1)-slice (which consists of all n-bit strings with exactly k+1 ones).

Yotam Dikstein, Irit Dinur, Yuval Filmus, and Prahladh Harsha. Boolean Function Analysis on High-Dimensional Expanders. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 38:1-38:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dikstein_et_al:LIPIcs.APPROX-RANDOM.2018.38, author = {Dikstein, Yotam and Dinur, Irit and Filmus, Yuval and Harsha, Prahladh}, title = {{Boolean Function Analysis on High-Dimensional Expanders}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {38:1--38:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.38}, URN = {urn:nbn:de:0030-drops-94421}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.38}, annote = {Keywords: high dimensional expanders, Boolean function analysis, sparse model} }

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

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

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

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@InProceedings{dinur_et_al:LIPIcs.ITCS.2017.37, author = {Dinur, Irit and Harsha, Prahladh and Venkat, Rakesh and Yuen, Henry}, title = {{Multiplayer Parallel Repetition for Expanding Games}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {37:1--37:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.37}, URN = {urn:nbn:de:0030-drops-81575}, doi = {10.4230/LIPIcs.ITCS.2017.37}, annote = {Keywords: Parallel Repetition, Multi-player, Expander} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

We study approximation of Boolean functions by low-degree polynomials over the ring Z/2^kZ. More precisely, given a Boolean function F:{0,1}^n -> {0,1}, define its k-lift to be F_k:{0,1}^n -> {0,2^(k-1)} by F_k(x) = 2^(k-F(x)) (mod 2^k). We consider the fractional agreement (which we refer to as \gamma_{d,k}(F)) of F_k with degree d polynomials from Z/2^kZ[x_1,..,x_n].
Our results are the following:
* Increasing k can help: We observe that as k increases, gamma_{d,k}(F) cannot decrease. We give two kinds of examples where gamma_{d,k}(F) actually increases. The first is an infinite family of functions F such that gamma_{2d,2}(F) - gamma_{3d-1,1}(F) >= Omega(1). The second is an infinite family of functions F such that gamma_{d,1}(F) <= 1/2+o(1) - as small as possible - but gamma_{d,3}(F) >= 1/2 + Omega(1).
* Increasing k doesn't always help: Adapting a proof of Green [Comput. Complexity, 9(1):16--38, 2000], we show that irrespective of the value of k, the Majority function Maj_n satisfies gamma_{d,k}(Maj_n) <= 1/2+ O(d)/sqrt{n}. In other words, polynomials over Z/2^kZ for large k do not approximate the majority function any better than polynomials over Z/2Z.
We observe that the model we study subsumes the model of non-classical polynomials, in the sense that proving bounds in our model implies bounds on the agreement of non-classical polynomials with Boolean functions. In particular, our results answer questions raised by Bhowmick and Lovett [In Proc. 30th Computational Complexity Conf., pages 72-87, 2015] that ask whether non-classical polynomials approximate Boolean functions better than classical polynomials of the same degree.

Abhishek Bhrushundi, Prahladh Harsha, and Srikanth Srinivasan. On Polynomial Approximations Over Z/2^kZ*. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 12:1-12:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bhrushundi_et_al:LIPIcs.STACS.2017.12, author = {Bhrushundi, Abhishek and Harsha, Prahladh and Srinivasan, Srikanth}, title = {{On Polynomial Approximations Over Z/2^kZ*}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {12:1--12:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.12}, URN = {urn:nbn:de:0030-drops-70212}, doi = {10.4230/LIPIcs.STACS.2017.12}, annote = {Keywords: Polynomials over rings, Approximation by polynomials, Boolean functions, Non-classical polynomials} }

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

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

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

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

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

We consider the following multiplication-based tests to check if a given function f: F^n_q -> F_q is the evaluation of a degree-d polynomial over F_q for q prime.
Test_{e,k}: Pick P_1,...,P_k independent random degree-e polynomials and accept iff the function f P_1 ... P_k is the evaluation of a degree-(d + ek) polynomial.
We prove the robust soundness of the above tests for large values of e, answering a question of Dinur and Guruswami (FOCS 2013). Previous soundness analyses of these tests were known only for the case when either e = 1 or k = 1. Even for the case k = 1 and e > 1, earlier soundness analyses were not robust.
We also analyze a derandomized version of this test, where (for example) the polynomials P_1 ,... , P_k can be the same random polynomial P. This generalizes a result of Guruswami et al. (STOC 2014).
One of the key ingredients that go into the proof of this robust soundness is an extension of the standard Schwartz-Zippel lemma over general finite fields F_q, which may be of independent interest.

Prahladh Harsha and Srikanth Srinivasan. Robust Multiplication-Based Tests for Reed-Muller Codes. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{harsha_et_al:LIPIcs.FSTTCS.2016.17, author = {Harsha, Prahladh and Srinivasan, Srikanth}, title = {{Robust Multiplication-Based Tests for Reed-Muller Codes}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {17:1--17:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.17}, URN = {urn:nbn:de:0030-drops-68524}, doi = {10.4230/LIPIcs.FSTTCS.2016.17}, annote = {Keywords: Polynomials over finite fields, Schwartz-Zippel lemma, Low degree testing, Low degree long code} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

We make progress on some questions related to polynomial approximations of AC^0. It is known, from the works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. 6th CCC 1991), that any AC^0 circuit of size s and depth d has an epsilon-error probabilistic polynomial over the reals of degree (log (s/epsilon))^{O(d)}. We improve this upper bound to (log s)^{O(d)}* log(1/epsilon), which is much better for small values of epsilon.
We give an application of this result by using it to resolve a question posed by Tal (ECCC 2014): we show that (log s)^{O(d)}* log(1/epsilon)-wise independence fools AC^0, improving on Tal's strengthening of Braverman's theorem (J. ACM 2010) that (log (s/epsilon))^{O(d)}-wise independence fools AC^0. Up to the constant implicit in the O(d), our result is tight. As far as we know, this is the first PRG construction for AC^0 that achieves optimal dependence on the error epsilon.
We also prove lower bounds on the best polynomial approximations to AC^0. We show that any polynomial approximating the OR function on n bits to a small constant error must have degree at least ~Omega(sqrt{log n}). This result improves exponentially on a recent lower bound demonstrated by Meka, Nguyen, and Vu (arXiv 2015).

Prahladh Harsha and Srikanth Srinivasan. On Polynomial Approximations to AC^0. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{harsha_et_al:LIPIcs.APPROX-RANDOM.2016.32, author = {Harsha, Prahladh and Srinivasan, Srikanth}, title = {{On Polynomial Approximations to AC^0}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {32:1--32:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.32}, URN = {urn:nbn:de:0030-drops-66550}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.32}, annote = {Keywords: Constant-depth Boolean circuits, Polynomials over reals, pseudo-random generators, k-wise independence} }

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

Let f: {0,1}^n*{0,1}^n -> {0,1} be a 2-party function. For every product distribution mu on {0,1}^n*{0,1}^n, we show that
CC^{mu}_{0.49}(f) = O(log(prt_{1/8}(f))*log(log(prt_{1/8}(f)))^2),
where CC^{mu}_{epsilon}(f) is the distributional communication complexity of f with error at most epsilon under the distribution mu and prt_{1/8}(f) is the partition bound of f, as defined by Jain and Klauck [Proc. 25th CCC, 2010]. We also prove a similar bound in terms of IC_{1/8}(f), the information complexity of f, namely,
CC^{mu}_{0.49}(f) = O((IC_{1/8}(f)*log(IC_{1/8}(f)))^2).
The latter bound was recently and independently established by Kol [Proc. 48th STOC, 2016] using a different technique.
We show a similar result for query complexity under product distributions. Let g: {0,1}^n -> {0,1} be a function. For every bit-wise product distribution mu on {0,1}^n, we show that
QC^{mu}_{0.49}(g) = O((log(qprt_{1/8}(g))*log(log(qprt_{1/8}(g))))^2),
where QC^{mu}_{epsilon}(g) is the distributional query complexity of f with error at most epsilon under the distribution mu and qprt_{1/8}(g) is the query partition bound of the function g.
Partition bounds were introduced (in both communication complexity and query complexity models) to provide LP-based lower bounds for randomized communication complexity and randomized query complexity. Our results demonstrate that these lower bounds are polynomially tight for product distributions.

Prahladh Harsha, Rahul Jain, and Jaikumar Radhakrishnan. Partition Bound Is Quadratically Tight for Product Distributions. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 135:1-135:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{harsha_et_al:LIPIcs.ICALP.2016.135, author = {Harsha, Prahladh and Jain, Rahul and Radhakrishnan, Jaikumar}, title = {{Partition Bound Is Quadratically Tight for Product Distributions}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {135:1--135:13}, 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.2016.135}, URN = {urn:nbn:de:0030-drops-62708}, doi = {10.4230/LIPIcs.ICALP.2016.135}, annote = {Keywords: partition bound, product distribution, communication complexity, query complexity} }

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Complete Volume

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

LIPIcs, Volume 45, FSTTCS'15, Complete Volume

35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@Proceedings{harsha_et_al:LIPIcs.FSTTCS.2015, title = {{LIPIcs, Volume 45, FSTTCS'15, Complete Volume }}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015}, URN = {urn:nbn:de:0030-drops-56671}, doi = {10.4230/LIPIcs.FSTTCS.2015}, annote = {Keywords: Software/Program Verification, Models of Computation, Modes of Computation, Complexity Measures and Classes, Nonnumerical Algorithms and Problems Specifying and Verifying and Reasoning about Programs, Mathematical Logic, Formal Languages} }

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Front Matter

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

Front Matter, Table of Contents, Preface, Conference Organization

35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. i-xiv, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{harsha_et_al:LIPIcs.FSTTCS.2015.i, author = {Harsha, Prahladh and Ramalingam, G.}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {i--xiv}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.i}, URN = {urn:nbn:de:0030-drops-56174}, doi = {10.4230/LIPIcs.FSTTCS.2015.i}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

We continue the study of covering complexity of constraint satisfaction problems (CSPs) initiated by Guruswami, Håstad and Sudan [SIAM J. Computing, 31(6):1663--1686, 2002] and Dinur and Kol [in Proc. 28th IEEE Conference on Computational Complexity, 2013]. The covering number of a CSP instance Phi, denoted by nu(Phi) is the smallest number of assignments to the variables of Phi, such that each constraint of Phi is satisfied by at least one of the assignments. We show the following results regarding how well efficient algorithms can approximate the covering number of a given CSP instance.
1. Assuming a covering unique games conjecture, introduced by Dinur and Kol, we show that for every non-odd predicate P over any constant sized alphabet and every integer K, it is NP-hard to distinguish between P-CSP instances (i.e., CSP instances where all the constraints are of type P) which are coverable by a constant number of assignments and those whose covering number is at least K. Previously, Dinur and Kol, using the same covering unique games conjecture, had shown a similar hardness result for every non-odd predicate over the Boolean alphabet that supports a pairwise independent distribution. Our generalization yields a complete characterization of CSPs over constant sized alphabet Sigma that are hard to cover since CSPs over odd predicates are trivially coverable with |Sigma| assignments.
2. For a large class of predicates that are contained in the 2k-LIN predicate, we show that it is quasi-NP-hard to distinguish between instances which have covering number at most two and covering number at least Omega(log(log(n))). This generalizes the 4-LIN result of Dinur and Kol that states it is quasi-NP-hard to distinguish between 4-LIN-CSP instances which have covering number at most two and covering number at least Omega(log(log(log(n)))).

Amey Bhangale, Prahladh Harsha, and Girish Varma. A Characterization of Hard-to-cover CSPs. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 280-303, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bhangale_et_al:LIPIcs.CCC.2015.280, author = {Bhangale, Amey and Harsha, Prahladh and Varma, Girish}, title = {{A Characterization of Hard-to-cover CSPs}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {280--303}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.280}, URN = {urn:nbn:de:0030-drops-50574}, doi = {10.4230/LIPIcs.CCC.2015.280}, annote = {Keywords: CSPs, Covering Problem, Hardness of Approximation, Unique Games, Invariance Principle} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

In this paper, we address the question of whether the recent derandomization results obtained by the use of the low-degree long code can be extended to other product settings. We consider two settings: (1) the graph product results of Alon, Dinur, Friedgut and Sudakov [GAFA, 2004] and (2) the "majority is stablest" type of result obtained by Dinur, Mossel and Regev [SICOMP, 2009] and Dinur and Shinkar [In Proc. APPROX, 2010] while studying the hardness of approximate graph coloring.
In our first result, we show that there exists a considerably smaller
subgraph of K_3^{\otimes R} which exhibits the following property
(shown for K_3^{\otimes R} by Alon et al.): independent sets close in
size to the maximum independent set are well approximated by dictators.
The "majority is stablest" type of result of Dinur et al. and Dinur
and Shinkar shows that if there exist two sets of vertices A and B
in K_3^{\otimes R} with very few edges with one endpoint in A and
another in B, then it must be the case that the two sets A and B
share a single influential coordinate. In our second result, we show
that a similar "majority is stablest" statement holds good for a
considerably smaller subgraph of K_3^{\otimes R}. Furthermore using
this result, we give a more efficient reduction from Unique Games
to the graph coloring problem, leading to improved hardness of
approximation results for coloring.

Irit Dinur, Prahladh Harsha, Srikanth Srinivasan, and Girish Varma. Derandomized Graph Product Results Using the Low Degree Long Code. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 275-287, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{dinur_et_al:LIPIcs.STACS.2015.275, author = {Dinur, Irit and Harsha, Prahladh and Srinivasan, Srikanth and Varma, Girish}, title = {{Derandomized Graph Product Results Using the Low Degree Long Code}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {275--287}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.275}, URN = {urn:nbn:de:0030-drops-49200}, doi = {10.4230/LIPIcs.STACS.2015.275}, annote = {Keywords: graph product, derandomization, low degree long code, graph coloring} }

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

The main result of this paper is an optimal strong direct
product result for the two-party public-coin randomized communication
complexity of the Tribes function. This is proved by providing an
alternate proof of the optimal lower bound of Omega(n) for the randomised communication complexity of the Tribes function using the so-called smooth-rectangle bound, introduced by Jain and Klauck [CCC/2010]. The optimal Omega(n) lower bound for Tribes was originally proved by Jayram, Kumar and Sivakumar [STOC/2003], using a more powerful lower bound technique, namely the information complexity bound. The information complexity bound is known to be at least as strong a lower bound method as the smooth-rectangle bound [Kerenidis et al, 2012]. On the other hand, we are not aware of any function or relation for which the smooth-rectangle bound is (asymptotically) smaller than its public-coin randomized communication complexity. The optimal direct product for Tribes is obtained by combining our smooth-rectangle bound for tribes with the strong direct product result of Jain and Yao (2012) in terms of smooth-rectangle bound.

Prahladh Harsha and Rahul Jain. A Strong Direct Product Theorem for the Tribes Function via the Smooth-Rectangle Bound. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 141-152, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{harsha_et_al:LIPIcs.FSTTCS.2013.141, author = {Harsha, Prahladh and Jain, Rahul}, title = {{A Strong Direct Product Theorem for the Tribes Function via the Smooth-Rectangle Bound}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {141--152}, 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.2013.141}, URN = {urn:nbn:de:0030-drops-43670}, doi = {10.4230/LIPIcs.FSTTCS.2013.141}, annote = {Keywords: Rectangle bound, Tribes function, Strong direct product} }

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