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Some Topics in Analysis of Boolean Functions
"... This article accompanies a tutorial talk given at the 40th ACM STOC conference. In it, we give a brief introduction to Fourier analysis of boolean functions and then discuss some applications: Arrow’s Theorem and other ideas from the theory of Social Choice; the BonamiBeckner Inequality as an exten ..."
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This article accompanies a tutorial talk given at the 40th ACM STOC conference. In it, we give a brief introduction to Fourier analysis of boolean functions and then discuss some applications: Arrow’s Theorem and other ideas from the theory of Social Choice; the BonamiBeckner Inequality as an extension of Chernoff/Hoeffding bounds to higherdegree polynomials; and, hardness for approximation algorithms.
New lower bounds for Approximation Algorithms in the LovaszSchrijver Hierarchy
, 2006
"... Determining how well we can efficiently compute approximate solutions to NPhard problems is of great theoretical and practical interest. Typically the famous PCP theorem is used for showing that a problem has no algorithms computing good approximations. Unfortunately, for many problem this approach ..."
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Determining how well we can efficiently compute approximate solutions to NPhard problems is of great theoretical and practical interest. Typically the famous PCP theorem is used for showing that a problem has no algorithms computing good approximations. Unfortunately, for many problem this approach has failed. Nevertheless, for such problems, we may instead be able to show that a large subclass of algorithms cannot compute good approximations. This thesis takes this approach, concentrating on subclasses of algorithms defined by the LS and LS+ LovászSchrijver hierarchies. These subclasses define hierarchies of algorithms where algorithms in higher levels (also called ”rounds”) require more time, but may compute better approximations. Algorithms in the LS hierarchy are based on linear programming relaxations while those in the more powerful LS+ hierarchy are based on semidefinite programming relaxations. Most known approximation algorithms lie within the first two–three levels of the LS+ hierarchy, including the recent celebrated approximation algorithms of Goemans
HARDNESS OF SOLVING SPARSE OVERDETERMINED LINEAR SYSTEMS: A 3QUERY PCP OVER INTEGERS
"... Abstract. A classic result due to H˚astad established that for every constant ε> 0, given an overdetermined system of linear equations over a finite field Fq where each equation depends on exactly 3 variables and at least a fraction (1 − ε) of the equations can be satisfied, it is NPhard to sati ..."
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Abstract. A classic result due to H˚astad established that for every constant ε> 0, given an overdetermined system of linear equations over a finite field Fq where each equation depends on exactly 3 variables and at least a fraction (1 − ε) of the equations can be satisfied, it is NPhard to satisfy even a fraction ` 1 q + ε ´ of the equations. In this work, we prove the analog of H˚astad’s result for equations over the integers (as well as the reals). Formally, we prove that for every ε, δ> 0, given a system of linear equations with integer coefficients where each equation is on 3 variables, it is NPhard to distinguish between the following two cases: (i) There is an assignment of integer values to the variables that satisfies at least a fraction (1 − ε) of the equations, and (ii) No assignment even of real values to the variables satisfies more than a fraction δ of the equations. 1.
ABSTRACT A 3Query PCP over Integers
"... A classic result due to H˚astad [11] established that for every constant ε> 0, given an overdetermined system of linear equations over a finite field Fq where each equation depends on exactly 3 variables and at least a fraction (1 − ε) of the equations “ can be ” satisfied, it is NPhard to satis ..."
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A classic result due to H˚astad [11] established that for every constant ε> 0, given an overdetermined system of linear equations over a finite field Fq where each equation depends on exactly 3 variables and at least a fraction (1 − ε) of the equations “ can be ” satisfied, it is NPhard to satisfy even a 1 fraction + ε of the equations. q In this work, we prove the analog of H˚astad’s result for equations over the integers (as well as the reals). Formally, we prove that for every ε, δ> 0, given a system of linear equations with integer coefficients where each equation is on 3 variables, it is NPhard to distinguish between the following two cases: (i) There is an assignment of integer values to the variables that satisfies at least a fraction (1−ε) of the equations, and (ii) No assignment even of real values to the variables satisfies more than a fraction δ of the equations.