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

A recent result of Moshkovitz [Moshkovitz14] presented an ingenious method to provide a completely elementary proof of the Parallel Repetition Theorem for certain projection games via a construction called fortification. However, the construction used in [Moshkovitz14] to fortify arbitrary label cover instances using an arbitrary extractor is insufficient to prove parallel repetition. In this paper, we provide a fix by using a stronger graph that we call fortifiers. Fortifiers are graphs that have both l_1 and l_2 guarantees on induced distributions from large subsets.
We then show that an expander with sufficient spectral gap, or a bi-regular extractor with stronger parameters (the latter is also the construction used in an independent update [Moshkovitz15] of [Moshkovitz14] with an alternate argument), is a good fortifier. We also show that using a fortifier (in particular l_2 guarantees) is necessary for obtaining the robustness required for fortification.

Amey Bhangale, Ramprasad Saptharishi, Girish Varma, and Rakesh Venkat. On Fortification of Projection Games. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 497-511, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bhangale_et_al:LIPIcs.APPROX-RANDOM.2015.497, author = {Bhangale, Amey and Saptharishi, Ramprasad and Varma, Girish and Venkat, Rakesh}, title = {{On Fortification of Projection Games}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {497--511}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.497}, URN = {urn:nbn:de:0030-drops-53204}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.497}, annote = {Keywords: Parallel Repetition, Fortification} }

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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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