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### Improved NP-Inapproximability for 2-Variable Linear Equations

Johan Håstad, Sangxia Huang, Rajsekar Manokaran, Ryan O'Donnell, John Wright, Marc Herbstritt
2015 International Workshop on Approximation Algorithms for Combinatorial Optimization
The well known constraint satisfaction problem (CSP) 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.  ...

### Improved NP-inapproximability for 2-Variable Linear Equations *

Johan Håstad, Sangxia Huang, Rajsekar Manokaran, Ryan O'donnell, John Wright
unpublished
An instance of the 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.  ...

### Better Inapproximability Results for MaxClique, Chromatic Number and Min-3Lin-Deletion [chapter]

Subhash Khot, Ashok Kumar Ponnuswami
2006 Lecture Notes in Computer Science
An instance of Min-3Lin-Deletion is a system of 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.  ...

### The inapproximability of lattice and coding problems with preprocessing

Uriel Feige, Daniele Micciancio
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  ...

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

Min-Zheng Shieh, Shi-Chun Tsai
2012 IEEE Transactions on Information Theory
any constant unless (quasi-polynomial time), and 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.  ...

### Approximation of barter exchanges with cycle length constraints [article]

Suiqian Luo, Pingzhong Tang, Chenggang Wu, Jianyang Zeng
2016 arXiv   pre-print
The problem has previously been shown to be 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.  ...

### Inapproximability of Combinatorial Optimization Problems [chapter]

Luca Trevisan
Acknowledgements I thank Vangelis Paschos 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]

Luca Trevisan
Acknowledgements I thank Vangelis Paschos 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

Miroslav Chlebík, Janka Chlebíková
2008 Theoretical Computer Science
Our 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  ...

### Inapproximability of Combinatorial Optimization Problems [article]

Luca Trevisan
2004 arXiv   pre-print
Acknowledgements I thank Vangelis Paschos 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∞)

WenBin Chen, Jiangtao Meng, Dengpan Yin
2008 Theoretical Computer Science
In this paper, we study the 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.  ...

### Approximation Hardness for Small Occurrence Instances of NP-Hard Problems [chapter]

Miroslav Chlebík, Janka Chlebíková
2003 Lecture Notes in Computer Science
New structural results which 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.  ...

### On the complexity of approximating k-set packing

Elad Hazan, Shmuel Safra, Oded Schwartz
2006 Computational Complexity
This 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

Subhash Khot
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 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.  ...

### Approximation Hardness of Graphic TSP on Cubic Graphs [article]

Marek Karpinski, Richard Schmied
2013 arXiv   pre-print
We prove explicit approximation hardness results 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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