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Improved NP-Inapproximability for 2-Variable Linear Equations

2015
*
International Workshop on Approximation Algorithms for Combinatorial Optimization
*

The well known constraint satisfaction problem (CSP)

doi:10.4230/lipics.approx-random.2015.341
dblp:conf/approx/HastadHMOW15
fatcat:7vx4k2vvangwzl6fijdpif2hfe
*2*-Lin(q) is defined as follows: Given n*variables*x 1 , . . . , x n , as well as a system of*equations*(constraints) of the form "*for*constants b ∈ ... Given such a system in which it's possible to satisfy all but an fraction of the*equations*, we show it is*NP*-hard to satisfy all but a C fraction of*equations*,*for*any C < 11 8 = 1.375 (and any 0 < ≤ 1 ... Acknowledgments The authors would like to warmly thank Per Austrin*for*his assistance with computer analysis of the 11 8 -gadget. ...##
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Improved NP-inapproximability for 2-Variable Linear Equations *

unpublished

An instance of the

fatcat:pmoawqe7yvfyblvbth553qvvim
*2*-Lin(*2*) problem is a system of*equations*of the form "x i + x j = b (mod*2*)". ... Given such a system in which it's possible to satisfy all but an fraction of the*equations*, we show it is*NP*-hard to satisfy all but a C fraction of the*equations*,*for*any C < 11 8 = 1.375 (and any 0 < ... The authors would like to warmly thank Per Austrin*for*his assistance with computer analysis of the 11 8 -gadget. ...##
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Better Inapproximability Results for MaxClique, Chromatic Number and Min-3Lin-Deletion
[chapter]

2006
*
Lecture Notes in Computer Science
*

An instance of Min-3Lin-Deletion is a system of

doi:10.1007/11786986_21
fatcat:yxhwyqsz2re2zith2h7a37wowa
*linear**equations*modulo*2*, where each*equation*is over three*variables*. ... We show that*for*any constant γ > 0, there is no polynomial time algorithm that approximates these problems within factor n/*2*(log n) 3/4+γ in an n vertex graph, assuming*NP*BPTIME(*2*(log n) O(1) ). ... Therefore, A 3 has N 3 = N O(1)*2**linear**equations*where each*equation*is over kO(log*2*N*2*) = O(log 3 N*2*)*variables*. ...##
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The inapproximability of lattice and coding problems with preprocessing

2004
*
Journal of computer and system sciences (Print)
*

*for*the nearest codeword problem with preprocessing (NCPP), proving that

*for*any finite field GFðqÞ; NCPP over GFðqÞ is

*NP*-hard to approximate within any factor less than 5 3 : r 2004 Published by Elsevier ... It follows that there are lattices

*for*which the closest vector problem cannot be approximated within factors go ffiffi 5 3 q in polynomial time, no matter how the lattice is represented, unless

*NP*is ...

*for*small alphabets, in particular

*for*q ¼

*2*; the

*inapproximability*factor is close to 4 3 : Then, in Section 3, we use standard techniques of concatenating codes [10] to

*improve*the

*inapproximability*...

##
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Inapproximability Results for the Weight Problems of Subgroup Permutation Codes

2012
*
IEEE Transactions on Information Theory
*

any constant unless (quasi-polynomial time), and

doi:10.1109/tit.2012.2208618
fatcat:enguqdn6kzdfplgvsd7w6xv4ti
*2*) there is no polynomial-time -approximation algorithm*for*the minimum weight problem*for*any constant unless . ... In this paper, we give*inapproximability*results*for*the minimum and maximum weight problems of subgroup permutation codes under several well-known metrics. ... Definition 7 ( 7 MAX-E3-LIN-*2*): Given a system of*linear**equations*over with exactly 3*variables*in each*equation*, determine the maximum number of*equations*which can be satisfied simultaneously. ...##
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Approximation of barter exchanges with cycle length constraints
[article]

2016
*
arXiv
*
pre-print

The problem has previously been shown to be

arXiv:1605.08863v1
fatcat:kwmmo3nxofgbrm4c4t522ez6ou
*NP*-hard. We advance the understanding of this problem by the following contributions. We prove three constant*inapproximability*results*for*this problem. ...*For*the unweighted graphs, we prove that this problem is*NP*-hard to approximate within a factor of 698/697. ... x 1 c 1 , c*2*, . . . , c m , where each constraint is a*linear**equation*modular*2*with 3*variables*. ...##
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Inapproximability of Combinatorial Optimization Problems
[chapter]

2014
*
Paradigms of Combinatorial Optimization
*

Acknowledgements I thank Vangelis Paschos

doi:10.1002/9781119005353.ch13
fatcat:o6srcissmrepxcci27ebredg34
*for*giving me the opportunity to write this survey, and*for*being very patient with my inability to meet deadlines. ... I am grateful to James Lee and Robi Krautghamer*for*their very valuable suggestions that gave shape to Section 7. ... over GF (*2*) with three*variables*per*equation*. ...##
###
Inapproximability of Combinatorial Optimization Problems
[chapter]

2013
*
Paradigms of Combinatorial Optimization
*

Acknowledgements I thank Vangelis Paschos

doi:10.1002/9781118600207.ch13
fatcat:xjl7lrlwmje3nguo6etwiuzxwi
*for*giving me the opportunity to write this survey, and*for*being very patient with my inability to meet deadlines. ... I am grateful to James Lee and Robi Krautghamer*for*their very valuable suggestions that gave shape to Section 7. ... over GF (*2*) with three*variables*per*equation*. ...##
###
The Steiner tree problem on graphs: Inapproximability results

2008
*
Theoretical Computer Science
*

Our

doi:10.1016/j.tcs.2008.06.046
fatcat:hw3lm5eg7nf5xgmw6wupbp7wnm
*inapproximability*results are stated in a parametric way, and explicit hardness factors would be*improved*automatically by providing gadgets and/or expanders with better parameters. ... We show that it is*NP*-hard to approximate the Steiner tree problem within a factor 96/95. ... Preliminaries Our*inapproximability*results use a reduction from Håstad's*NP*-hard gap type result*for*Max-E3-Lin-*2*, the maximum satisfiability problem*for**linear**equations*modulo*2*with exactly 3 unknowns ...##
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Inapproximability of Combinatorial Optimization Problems
[article]

2004
*
arXiv
*
pre-print

Acknowledgements I thank Vangelis Paschos

arXiv:cs/0409043v1
fatcat:tgyatqut7fen3etghmem66kk5m
*for*giving me the opportunity to write this survey, and*for*being very patient with my inability to meet deadlines. ... I am grateful to James Lee and Robi Krautghamer*for*their very valuable suggestions that gave shape to Section 7. ... over GF (*2*) with three*variables*per*equation*. ...##
###
An improved lower bound for approximating minimum GCD multiplier in ℓ∞ norm (GCDM∞)

2008
*
Theoretical Computer Science
*

In this paper, we study the

doi:10.1016/j.tcs.2007.09.030
fatcat:3rlduk6bo5bz7g7fqfhfmaong4
*inapproximability*of the following*NP*-complete number theoretic optimization problems introduced by Rössner and Seifert [C. Rössner, J.P. ... Seifert, The complexity of approximate optima*for*greatest common divisor computations, in: ... Acknowledgments The authors would like to thank the anonymous referees*for*their careful reading of the manuscript and many useful suggestions. ...##
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Approximation Hardness for Small Occurrence Instances of NP-Hard Problems
[chapter]

2003
*
Lecture Notes in Computer Science
*

New structural results which

doi:10.1007/3-540-44849-7_21
fatcat:dvffpjwhvvg6lot3sstds2ffaq
*improve*the known bounds*for*3-regular amplifiers and hence the*inapproximability*results*for*numerous small occurrence problems studied earlier by Berman and Karpinski are ... We present parametrized reductions*for*some packing and covering problems, including 3-Dimensional Matching, and prove the best known*inapproximability*results even*for*highly restricted versions of them ... Amplifier parametrized known reductions We call HYBRID a system of*linear**equations*over Z*2*, each*equation*either with*2*or with 3*variables*. ...##
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On the complexity of approximating k-set packing

2006
*
Computational Complexity
*

This

doi:10.1007/s00037-006-0205-6
fatcat:3hemtqt4hrbtzjzwidhlhsqota
*improves*the previous factor of A similar technique yields*NP*-hardness factors of 54 53 − ε, 30 29 − ε and 23 22 − ε*for*4-SP, 5-SP and 6-SP respectively. ... Definition 3 (*Linear**Equations*). MAX-3-LIN-q is the following optimization problem: Input: A set Φ of*linear**equations*over GF (q), each depending on 3*variables*. ... We*improve*the*inapproximability*factor*for*the variant k-DM, and show: Theorem 1. ...##
###
Inapproximability of NP-complete Problems, Discrete Fourier Analysis, and Geometry

2011
*
Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)
*

This article gives a survey of recent results that connect three areas in computer science and mathematics: (1) (Hardness of) computing approximate solutions to

doi:10.1142/9789814324359_0163
fatcat:64vr5c2f6fbfnosflfpff2t5mu
*NP*-complete problems. (*2*) Fourier analysis ... Max-3Lin and*Linearity*Test with Perturbation Max-3Lin Problem: Given a system of*linear**equations*over GF (*2*) with each*equation*containing three*variables*. ... We are given a system of*linear**equations*over GF (*2*) with three*variables*in each*equation*and the goal is to find an assignment that satisfies the maximum number of*equations*. ...##
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Approximation Hardness of Graphic TSP on Cubic Graphs
[article]

2013
*
arXiv
*
pre-print

We prove explicit approximation hardness results

arXiv:1304.6800v2
fatcat:3pgb67q6prg3deqg7cr65qqmsq
*for*the Graphic TSP on cubic and subcubic graphs as well as the new*inapproximability*bounds*for*the corresponding instances of the (1,2)-TSP. ... The proof technique uses new modular constructions of simulating gadgets*for*the restricted cubic and subcubic instances. ... Acknowledgments We thank Leen Stougie and Ola Svensson*for*a number of interesting discussions. ...
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