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The Projection Games Conjecture and the Hardness of Approximation of SSAT and related problems [article]

Priyanka Mukhopadhyay
2019 arXiv   pre-print
The Super-SAT or SSAT problem was introduced by Dinur, Kindler, Raz and Safra[2002,2003] to prove the NP-hardness of approximation of two popular lattice problems - Shortest Vector Problem (SVP) and Closest  ...  This implies hardness of approximation of SVP and CVP within polynomial factors, assuming the Projection Games Conjecture.  ...  The author would like to thank Dana Moshkovitz for clarifying some concepts about Projection Games Conjecture, via personal correspondence with Divesh Aggarwal.  ... 
arXiv:1907.05548v2 fatcat:d4kh6fqsr5eglai333xy2uiauq

Fixed Parameter Inapproximability for Clique and SetCover in Time Super-exponential in OPT [article]

Mohammad T. Hajiaghayi and Rohit Khandekar and Guy Kortsarz
2013 arXiv   pre-print
In this paper, we consider proving inapproximability in terms of OPT and thus we base the foundations of fixed parameter inapproximability.  ...  Another conjecture used is Projection Game Conjecture (pgc) due to Moshkovitz [21] .  ...  Projection Game Conjecture (pgc) There exists some constant c and pcp of size m · ρ(m) · poly(1/ǫ), with soundness 1/ǫ for any ǫ so that 1/ǫ ≤ m c .  ... 
arXiv:1310.2711v2 fatcat:h4ujlrnotjeshfpeqhovm6vuwm

On subexponential running times for approximating directed Steiner tree and related problems [article]

Marek Cygan, Guy Kortsarz, Bundit Laekhanukit
2018 arXiv   pre-print
This paper concerns proving almost tight (super-polynomial) running times, for achieving desired approximation ratios for various problems.  ...  Our result follows by analyzing the work of Halperin and Krauthgamer [STOC, 2003]. The same lower and upper bounds hold for CST.  ...  The work of Marek Cygan is part of a project TOTAL that has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement  ... 
arXiv:1811.00710v1 fatcat:ex3gvufbo5hvhk2cmo5egbf4rq

Parallel Repetition from Fortification

Dana Moshkovitz
2014 2014 IEEE 55th Annual Symposium on Foundations of Computer Science  
The latter can be used for hardness of approximation as in the work of Håstad. (2) Starting from the work of the author and Raz, we get a projection PCP theorem with the smallest soundness error known  ...  This connection could shed light on the so-called Projection Games Conjecture, which asks for projection PCP with minimal error.  ...  ACKNOWLEDGEMENTS I am thankful to Ran Raz for discussions, and to Henry Yuen and an anonymous reviewer for a careful reading of the paper.  ... 
doi:10.1109/focs.2014.51 dblp:conf/focs/Moshkovitz14 fatcat:xrjhuuhx7nbpjbu6iwwr4oayi4

The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 18231)

Martin Grohe, Venkatesan Guruswami, Stanislav Zivny, Michael Wagner
2018 Dagstuhl Reports  
CSPs constitute a very rich and yet sufficiently manageable class of problems to give a good perspective on general computational phenomena.  ...  For instance, they help to understand which mathematical properties make a computational problem tractable (in a wide sense, e.g., polynomial-time solvable, non-trivially approximable, fixed-parameter  ...  -The Constraint Satisfaction Problem: Complexity and Approximability From Weak to Strong LP Gaps for all CSPs We study the approximability of constraint satisfaction problems (CSPs) by linear programming  ... 
doi:10.4230/dagrep.8.6.1 dblp:journals/dagstuhl-reports/GroheGZ18 fatcat:3bqo62ly3rgzlnh3bmkvwbuwea

Variable Selection is Hard [article]

Dean Foster, Howard Karloff, Justin Thaler
2014 arXiv   pre-print
Variable selection for sparse linear regression is the problem of finding, given an m x p matrix B and a target vector y, a sparse vector x such that Bx approximately equals y.  ...  We prove a similar result for a statistical version of the problem in which the data are corrupted by noise.  ...  This is the key to establishing hardness for super-logarithmic approximation ratios, and to obtaining hardness results even when we only require an approximate solution to the system of linear equations  ... 
arXiv:1412.4832v1 fatcat:m2novipbpjcnfh2phnjkvgayxu

Pseudorandomness and the Minimum Circuit Size Problem

Rahul Santhanam, Michael Wagner
2020 Innovations in Theoretical Computer Science  
We also show that for a certain natural variant of MCSP, there is a polynomial-time reduction from approximating the problem well in the worst case to solving it on average.  ...  We explore the possibility of basing one-way functions on the average-case hardness of the fundamental Minimum Circuit Size Problem (MCSP[s]), which asks whether a Boolean function on n bits specified  ...  The analysis of the game uses the approximate Min-Max theorem of [9, 38] and is fairly general.  ... 
doi:10.4230/lipics.itcs.2020.68 dblp:conf/innovations/Santhanam20 fatcat:pgatorc5k5hlrknytijhfgjufy

Limits of Approximation Algorithms: PCPs and Unique Games (DIMACS Tutorial Lecture Notes) [article]

Prahladh Harsha, Moses Charikar, Matthew Andrews, Sanjeev Arora, Subhash Khot, Dana Moshkovitz, Lisa Zhang, Ashkan Aazami, Dev Desai, Igor Gorodezky, Geetha Jagannathan, Alexander S. Kulikov, Darakhshan J. Mir, Alantha Newman (+3 others)
2010 arXiv   pre-print
This tutorial was jointly sponsored by the DIMACS Special Focus on Hardness of Approximation, the DIMACS Special Focus on Algorithmic Foundations of the Internet, and the Center for Computational Intractability  ...  These are the lecture notes for the DIMACS Tutorial "Limits of Approximation Algorithms: PCPs and Unique Games" held at the DIMACS Center, CoRE Building, Rutgers University on 20-21 July, 2009.  ...  To handle such problems, Khot formulated the unique games conjecture [Kho02] . This conjecture postulates the hardness of unique label cover, where the projections π e on the edges are permutations.  ... 
arXiv:1002.3864v1 fatcat:ys2sgp7aqvgmtio32575q3e43y

Nonlocal Games, Compression Theorems, and the Arithmetical Hierarchy [article]

Hamoon Mousavi, Seyed Sajjad Nezhadi, Henry Yuen
2021 arXiv   pre-print
We investigate the connection between the complexity of nonlocal games and the arithmetical hierarchy, a classification of languages according to the complexity of arithmetical formulas defining them.  ...  We prove that deciding whether the quantum value of a two-player nonlocal game is exactly equal to 1 is complete for Π_2; this class is in the second level of the arithmetical hierarchy and corresponds  ...  H.M. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC).  ... 
arXiv:2110.04651v2 fatcat:hqlhans4xzfoldgo6jubudbkgy

Tight FPT Approximations for k-Median and k-Means

Vincent Cohen-Addad, Anupam Gupta, Amit Kumar, Euiwoong Lee, Jason Li, Michael Wagner
2019 International Colloquium on Automata, Languages and Programming  
We investigate the fine-grained complexity of approximating the classical k-Median/k-Means clustering problems in general metric spaces.  ...  We show how to improve the approximation factors to (1 + 2/e + ε) and (1 + 8/e + ε) respectively, using algorithms that run in fixed-parameter time.  ...  To do this, we construct a variable-clause game from a 3-SAT instance, merge clause vertices into super-vertices, and then use r rounds of parallel repetition.  ... 
doi:10.4230/lipics.icalp.2019.42 dblp:conf/icalp/Cohen-AddadG0LL19 fatcat:4w3h3lmsbjfcfnoge7kru67pgi

Time-Approximation Trade-offs for Inapproximable Problems [article]

Édouard Bonnet, Michael Lampis, Vangelis Th. Paschos
2015 arXiv   pre-print
We observe that if such ratios could be achieved in polynomial time, the ETH or the Projection Games Conjecture would fail.  ...  We tackle a number of problems: For Min Independent Dominating Set, Max Induced Path, Forest and Tree, for any r(n), a simple, known scheme gives an approximation ratio of r in time roughly r^n/r.  ...  We also observe that, if the ETH and the Projection Games Conjecture [21] are true, there exists c > 0 such that m c -approximation cannot be achieved in polynomial time.  ... 
arXiv:1502.05828v1 fatcat:j3yrrqa7srcvpdctoxj3vrxepm

Time-approximation trade-offs for inapproximable problems

Édouard Bonnet, Michael Lampis, Vangelis Th. Paschos
2018 Journal of computer and system sciences (Print)  
The projection games conjecture and the NP-hardness of ln napproximating set-cover. In A.  ...  We observe that if such ratios could be achieved in polynomial time, the ETH or the Projection Games Conjecture would fail. 21 Dana Moshkovitz.  ...  We also observe that, if the ETH and the Projection Games Conjecture [21] are true, there exists c > 0 such that m c -approximation cannot be achieved in polynomial time.  ... 
doi:10.1016/j.jcss.2017.09.009 fatcat:owftfddbcvejvlsb72epihnxfq

Proof Complexity (Dagstuhl Seminar 18051)

Albert Atserias, Jakob Nordström, Pavel Pudlák, Rahul Santhanam, Michael Wagner
2018 Dagstuhl Reports  
Moreover, it allows to analyse the power and limitations of satisfiability algorithms (SAT solvers) used in industrial applications with formulas containing up to millions of variables.  ...  NP problem, and in the ensuing decades many powerful techniques have been discovered for analyzing different proof systems.  ...  Jan Krajíček, Pavel Pudlák, and Alan R. Woods. An exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle. Random Structures and Algorithms, 7(1):15-40, 1995.  ... 
doi:10.4230/dagrep.8.1.124 dblp:journals/dagstuhl-reports/AtseriasNPS18 fatcat:5ksfbo2ehfhspcbuw4ppcuyaqu

On the optimality of semidefinite relaxations for average-case and generalized constraint satisfaction

Boaz Barak, Guy Kindler, David Steurer
2013 Proceedings of the 4th conference on Innovations in Theoretical Computer Science - ITCS '13  
hardness for the densest k subgraph problem and hard instances for the Sliding Scale Conjecture of Bellare, Goldwasser, Lund and Russell (1993) .  ...  We provide several types of indirect evidence for the truth of this hypothesis, and also show that it (and its variants) imply several conjectures in hardness of approximation including polynomial factor  ...  One can view the classic result of Håstad [Hås01] as some hardness of approximation on this form, for the case of predicates such as k-XOR and k-SAT.  ... 
doi:10.1145/2422436.2422460 dblp:conf/innovations/BarakKS13 fatcat:k3omfqtzq5h7hgx5p3qpzabgpi

Affine reductions for LPs and SDPs [article]

Gábor Braun, Sebastian Pokutta, Daniel Zink
2016 arXiv   pre-print
In the case of SDPs, we obtain inapproximability results for these problems relative to the SDP-inapproximability of MaxCUT.  ...  We define a reduction mechanism for LP and SDP formulations that degrades approximation factors in a controlled fashion.  ...  We are indebted to Siu On Chan for some of the PCP inapproximability bounds as well as Santosh Vempala for the helpful discussions.  ... 
arXiv:1410.8816v5 fatcat:tqklld6tnbbmvokxxag2l6sjem
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