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

We show improved monotonicity testers for the Boolean hypercube under the p-biased measure, as well as over the hypergrid [m]ⁿ. Our results are:
1) For any p ∈ (0,1), for the p-biased hypercube we show a non-adaptive tester that makes Õ(√n/ε²) queries, accepts monotone functions with probability 1 and rejects functions that are ε-far from monotone with probability at least 2/3.
2) For all m ∈ ℕ, we show an Õ(√nm³/ε²) query monotonicity tester over [m]ⁿ. We also establish corresponding directed isoperimetric inequalities in these domains, analogous to the isoperimetric inequality in [Subhash Khot et al., 2018]. Previously, the best known tester due to Black, Chakrabarty and Seshadhri [Hadley Black et al., 2018] had Ω(n^{5/6}) query complexity. Our results are optimal up to poly-logarithmic factors and the dependency on m.
Our proof uses a notion of monotone embeddings of measures into the Boolean hypercube that can be used to reduce the problem of monotonicity testing over an arbitrary product domains to the Boolean cube. The embedding maps a function over a product domain of dimension n into a function over a Boolean cube of a larger dimension n', while preserving its distance from being monotone; an embedding is considered efficient if n' is not much larger than n, and we show how to construct efficient embeddings in the above mentioned settings.

Mark Braverman, Subhash Khot, Guy Kindler, and Dor Minzer. Improved Monotonicity Testers via Hypercube Embeddings. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 25:1-25:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{braverman_et_al:LIPIcs.ITCS.2023.25, author = {Braverman, Mark and Khot, Subhash and Kindler, Guy and Minzer, Dor}, title = {{Improved Monotonicity Testers via Hypercube Embeddings}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {25:1--25:24}, 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.2023.25}, URN = {urn:nbn:de:0030-drops-175285}, doi = {10.4230/LIPIcs.ITCS.2023.25}, annote = {Keywords: Property Testing, Monotonicity Testing, Isoperimetric Inequalities} }

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

The k-Clique problem is a canonical hard problem in parameterized complexity. In this paper, we study the parameterized complexity of approximating the k-Clique problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a clique of size at least k/F(k) whenever the graph G has a clique of size k. When such an algorithm runs in time T(k) ⋅ poly(n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for the k-Clique problem.
Although, the non-existence of an F(k)-FPT-approximation algorithm for any computable sublinear function F is known under gap-ETH [Chalermsook et al., FOCS 2017], it has remained a long standing open problem to prove the same inapproximability result under the more standard and weaker assumption, W[1]≠FPT.
In a recent breakthrough, Lin [STOC 2021] ruled out constant factor (i.e., F(k) = O(1)) FPT-approximation algorithms under W[1]≠FPT. In this paper, we improve this inapproximability result (under the same assumption) to rule out every F(k) = k^{1/H(k)} factor FPT-approximation algorithm for any increasing computable function H (for example H(k) = log^∗ k).
Our main technical contribution is introducing list decoding of Hadamard codes over large prime fields into the proof framework of Lin.

Karthik C. S. and Subhash Khot. Almost Polynomial Factor Inapproximability for Parameterized k-Clique. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{karthikc.s._et_al:LIPIcs.CCC.2022.6, author = {Karthik C. S. and Khot, Subhash}, title = {{Almost Polynomial Factor Inapproximability for Parameterized k-Clique}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {6:1--6:21}, 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.2022.6}, URN = {urn:nbn:de:0030-drops-165680}, doi = {10.4230/LIPIcs.CCC.2022.6}, annote = {Keywords: Parameterized Complexity, k-clique, Hardness of Approximation} }

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

We give alternate proofs for three related results in analysis of Boolean functions, namely the KKL Theorem, Friedgut’s Junta Theorem, and Talagrand’s strengthening of the KKL Theorem. We follow a new approach: looking at the first Fourier level of the function after a suitable random restriction and applying the Log-Sobolev inequality appropriately. In particular, we avoid using the hypercontractive inequality that is common to the original proofs. Our proofs might serve as an alternate, uniform exposition to these theorems and the techniques might benefit further research.

Esty Kelman, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Theorems of KKL, Friedgut, and Talagrand via Random Restrictions and Log-Sobolev Inequality. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kelman_et_al:LIPIcs.ITCS.2021.26, author = {Kelman, Esty and Khot, Subhash and Kindler, Guy and Minzer, Dor and Safra, Muli}, title = {{Theorems of KKL, Friedgut, and Talagrand via Random Restrictions and Log-Sobolev Inequality}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {26:1--26:17}, 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.2021.26}, URN = {urn:nbn:de:0030-drops-135657}, doi = {10.4230/LIPIcs.ITCS.2021.26}, annote = {Keywords: Fourier Analysis, Hypercontractivity, Log-Sobolev Inequality} }

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

We propose a variant of the 2-to-1 Games Conjecture that we call the Rich 2-to-1 Games Conjecture and show that it is equivalent to the Unique Games Conjecture. We are motivated by two considerations. Firstly, in light of the recent proof of the 2-to-1 Games Conjecture [Subhash Khot et al., 2017; Irit Dinur et al., 2018; Irit Dinur et al., 2018; Subhash Khot et al., 2018], we hope to understand how one might make further progress towards a proof of the Unique Games Conjecture. Secondly, the new variant along with perfect completeness in addition, might imply hardness of approximation results that necessarily require perfect completeness and (hence) are not implied by the Unique Games Conjecture.

Mark Braverman, Subhash Khot, and Dor Minzer. On Rich 2-to-1 Games. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 27:1-27:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{braverman_et_al:LIPIcs.ITCS.2021.27, author = {Braverman, Mark and Khot, Subhash and Minzer, Dor}, title = {{On Rich 2-to-1 Games}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {27:1--27:20}, 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.2021.27}, URN = {urn:nbn:de:0030-drops-135666}, doi = {10.4230/LIPIcs.ITCS.2021.27}, annote = {Keywords: PCP, Unique-Games, Perfect Completeness} }

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**Published in:** LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

A systematic study of simultaneous optimization of constraint satisfaction problems was initiated by Bhangale et al. [ICALP, 2015]. The simplest such problem is the simultaneous Max-Cut. Bhangale et al. [SODA, 2018] gave a .878-minimum approximation algorithm for simultaneous Max-Cut which is almost optimal assuming the Unique Games Conjecture (UGC). For single instance Max-Cut, Goemans-Williamson [JACM, 1995] gave an α_GW-approximation algorithm where α_GW ≈ .87856720... which is optimal assuming the UGC.
It was left open whether one can achieve an α_GW-minimum approximation algorithm for simultaneous Max-Cut. We answer the question by showing that there exists an absolute constant ε₀ ≥ 10^{-5} such that it is NP-hard to get an (α_GW- ε₀)-minimum approximation for simultaneous Max-Cut assuming the Unique Games Conjecture.

Amey Bhangale and Subhash Khot. Simultaneous Max-Cut Is Harder to Approximate Than Max-Cut. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bhangale_et_al:LIPIcs.CCC.2020.9, author = {Bhangale, Amey and Khot, Subhash}, title = {{Simultaneous Max-Cut Is Harder to Approximate Than Max-Cut}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {9:1--9:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-156-6}, ISSN = {1868-8969}, year = {2020}, volume = {169}, editor = {Saraf, Shubhangi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.9}, URN = {urn:nbn:de:0030-drops-125610}, doi = {10.4230/LIPIcs.CCC.2020.9}, annote = {Keywords: Simultaneous CSPs, Unique Games hardness, Max-Cut} }

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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} }

Document

**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

The 2-to-2 Games Theorem of [Subhash Khot et al., 2017; Dinur et al., 2018; Dinur et al., 2018; Dinur et al., 2018] implies that it is NP-hard to distinguish between Unique Games instances with assignment satisfying at least (1/2-epsilon) fraction of the constraints vs. no assignment satisfying more than epsilon fraction of the constraints, for every constant epsilon>0. We show that the reduction can be transformed in a non-trivial way to give a stronger guarantee in the completeness case: For at least (1/2-epsilon) fraction of the vertices on one side, all the constraints associated with them in the Unique Games instance can be satisfied.
We use this guarantee to convert the known UG-hardness results to NP-hardness. We show:
1) Tight inapproximability of approximating independent sets in degree d graphs within a factor of Omega(d/(log^2 d)), where d is a constant.
2) NP-hardness of approximate the Maximum Acyclic Subgraph problem within a factor of 2/3+epsilon, improving the previous ratio of 14/15+epsilon by Austrin et al. [Austrin et al., 2015].
3) For any predicate P^{-1}(1) subseteq [q]^k supporting a balanced pairwise independent distribution, given a P-CSP instance with value at least 1/2-epsilon, it is NP-hard to satisfy more than (|P^{-1}(1)|/(q^k))+epsilon fraction of constraints.

Amey Bhangale and Subhash Khot. UG-Hardness to NP-Hardness by Losing Half. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bhangale_et_al:LIPIcs.CCC.2019.3, author = {Bhangale, Amey and Khot, Subhash}, title = {{UG-Hardness to NP-Hardness by Losing Half}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {3:1--3:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.3}, URN = {urn:nbn:de:0030-drops-108258}, doi = {10.4230/LIPIcs.CCC.2019.3}, annote = {Keywords: NP-hardness, Inapproximability, Unique Games Conjecture} }

Document

**Published in:** LIPIcs, Volume 93, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)

A Boolean function f:{0,1}^n\->{0,1} is called a dictator if it depends on exactly one variable i.e f(x_1, x_2, ..., x_n) = x_i for some i in [n]. In this work, we study a k-query dictatorship test. Dictatorship tests are central in proving many hardness results for constraint satisfaction problems.
The dictatorship test is said to have perfect completeness if it accepts any dictator function. The soundness of a test is the maximum probability with which it accepts any function far from a dictator. Our main result is a k-query dictatorship test with perfect completeness and soundness (2k + 1)/(2^k), where k is of the form 2^t -1 for any integer t > 2. This improves upon the result of [Tamaki-Yoshida, Random Structures & Algorithms, 2015] which gave a dictatorship test with soundness (2k + 3)/(2^k).

Amey Bhangale, Subhash Khot, and Devanathan Thiruvenkatachari. An Improved Dictatorship Test with Perfect Completeness. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 15:1-15:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bhangale_et_al:LIPIcs.FSTTCS.2017.15, author = {Bhangale, Amey and Khot, Subhash and Thiruvenkatachari, Devanathan}, title = {{An Improved Dictatorship Test with Perfect Completeness}}, booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)}, pages = {15:1--15:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-055-2}, ISSN = {1868-8969}, year = {2018}, volume = {93}, editor = {Lokam, Satya and Ramanujam, R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.15}, URN = {urn:nbn:de:0030-drops-83854}, doi = {10.4230/LIPIcs.FSTTCS.2017.15}, annote = {Keywords: Property Testing, Distatorship Test, Fourier Analysis} }

Document

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

We present an adaptive tester for the unateness property of Boolean functions. Given a function f:{0,1}^n -> {0,1} the tester makes O(n log(n)/epsilon) adaptive queries to the function. The tester always accepts a unate function, and rejects with probability at least 0.9 if a function is epsilon-far from being unate.

Subhash Khot and Igor Shinkar. An ~O(n) Queries Adaptive Tester for Unateness. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 37:1-37:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{khot_et_al:LIPIcs.APPROX-RANDOM.2016.37, author = {Khot, Subhash and Shinkar, Igor}, title = {{An \textasciitildeO(n) Queries Adaptive Tester for Unateness}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {37:1--37:7}, 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.2016.37}, URN = {urn:nbn:de:0030-drops-66603}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.37}, annote = {Keywords: property testing, boolean functions, unateness} }

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Invited Talk

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

The talk will present connections between approximability of NP-complete problems, analysis, and geometry, and the role played by the Unique Games Conjecture in facilitating these connections.

Subhash Khot. Hardness of Approximation (Invited Talk). In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{khot:LIPIcs.ICALP.2016.3, author = {Khot, Subhash}, title = {{Hardness of Approximation}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {3:1--3:1}, 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.2016.3}, URN = {urn:nbn:de:0030-drops-63395}, doi = {10.4230/LIPIcs.ICALP.2016.3}, annote = {Keywords: NP-completeness, Approximation algorithms, Inapproximability, Probabilistically Checkable Proofs, Discrete Fourier analysis} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

We study the natural problem of estimating the expansion of subsets of vertices on one side of a bipartite graph. More precisely, given a bipartite graph G(U,V,E) and a parameter beta, the goal is to find a subset V' subseteq V containing beta fraction of the vertices of V which minimizes the size of N(V'), the neighborhood of V'. This problem, which we call Bipartite Expansion, is a special case of submodular minimization subject to a cardinality constraint, and is also related to other problems in graph partitioning and expansion. Previous to this work, there was no hardness of approximation known for Bipartite Expansion.
In this paper we show the following strong inapproximability for Bipartite Expansion: for any constants tau, gamma > 0
there is no algorithm which, given a constant beta > 0 and a bipartite graph G(U,V,E), runs in polynomial time and decides whether
- (YES case) There is a subset S^* subseteq V s.t. |S^*| >= beta*|V| satisfying |N(S^*)| <= gamma |U|, or
- (NO case) Any subset S subseteq V s.t. |S| >= tau*beta*|V| satisfies |N(S)| >= (1 - gamma)|U|, unless
NP subseteq intersect_{epsilon > 0}{DTIME}(2^{n^epsi;on}) i.e. NP has subexponential time algorithms.
We note that our hardness result stated above is a vertex expansion analogue of the Small Set (Edge) Expansion Conjecture of
Raghavendra and Steurer 2010.

Subhash Khot and Rishi Saket. Hardness of Bipartite Expansion. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 55:1-55:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{khot_et_al:LIPIcs.ESA.2016.55, author = {Khot, Subhash and Saket, Rishi}, title = {{Hardness of Bipartite Expansion}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {55:1--55:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.55}, URN = {urn:nbn:de:0030-drops-63971}, doi = {10.4230/LIPIcs.ESA.2016.55}, annote = {Keywords: inapproximability, bipartite expansion, PCP, submodular minimization} }

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Invited Talk

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

Constraint satisfaction problems are some of the most well-studied NP-hard problems, 3SAT being a prominent example. It is known by Hastad's 1997 result that 3SAT is "approximation resistant" in the following sense: given a near-satisfiable instance, a trivial algorithm that assigns random boolean values to the variables satisfies 7/8 fraction of the constraints and no efficient algorithm can do strictly better unless P=NP!
3SAT is a CSP that corresponds to the ternary OR predicate. In general, a CSP has constraints given by some fixed predicate P:{0,1}^k -> {True, False} (on possibly negated variables) and the predicate is called approximation resistant if, on a near-satisfiable instance, it is computationally hard to perform strictly better than a random assignment.
The quest to understand approximation resistance has played a central role in the theory of probabilistically checkable proofs (PCPs) and hardness of approximation. This talk will give a survey of the topic, including recent work giving a complete characterization of approximation resistance (i.e. a necessary and sufficient condition on the predicate that makes the corresponding CSP approximation resistant).

Subhash Khot. On Approximation Resistance of Predicates (Invited Talk). In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, p. 19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{khot:LIPIcs.FSTTCS.2013.19, author = {Khot, Subhash}, title = {{On Approximation Resistance of Predicates}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {19--19}, 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.2013.19}, URN = {urn:nbn:de:0030-drops-44011}, doi = {10.4230/LIPIcs.FSTTCS.2013.19}, annote = {Keywords: Approximation resistance, Hardness of approximation, Probabilistically checkable proofs, Constraint satisfaction problem} }