A Structural Approach to Subset-Sum Problems

Vu, Van

doi:10.1007/978-3-540-85221-6_19

Cited by 5 publications

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“…, w n , see e.g. [Vu08,TV08]. Can corresponding extensions of Lemma 26 be established, proving that every threshold function admits a representation with weights that have the required structure?…”

Section: Discussionmentioning

confidence: 99%

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Improved Approximation of Linear Threshold Functions

Diakonikolas

Servedio

2012

comput. complex.

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We prove two main results on how arbitrary linear threshold functions f (x) = sign(w · x − θ) over the n-dimensional Boolean hypercube can be approximated by simple threshold functions.Our first result shows that every n-variable threshold function f is -close to a threshold function depending only on Inf(f ) 2 · poly(1/ ) many variables, where Inf(f ) denotes the total influence or average sensitivity of f. This is an exponential sharpening of Friedgut's well-known theorem [Fri98], which states that every Boolean function f is -close to a function depending only on 2 O(Inf(f )/ ) many variables, for the case of threshold functions. We complement this upper bound by showing that Ω(Inf(f ) 2 + 1/ 2 ) many variables are required for -approximating threshold functions.Our second result is a proof that every n-variable threshold function is -close to a threshold function with integer weights at most poly(n) · 2Õ (1/ 2/3 ) . This is an improvement, in the dependence on the error parameter , on an earlier result of [Ser07] which gave a poly(n) · 2Õ (1/ 2 ) bound. Our improvement is obtained via a new proof technique that uses strong anti-concentration bounds from probability theory. The new technique also gives a simple and modular proof of the original [Ser07] result, and extends to give low-weight approximators for threshold functions under a range of probability distributions other than the uniform distribution.

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Section: Discussionmentioning

confidence: 99%

“…Roughly speaking, if one forbids more and more additive structure in the w i 's, then one gets better and better anti-concentration; see e.g [Vu08,TV08]. and Chapter 7 of[TV06].…”

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confidence: 99%

Improved Approximation of Linear Threshold Functions

Diakonikolas

Servedio

2012

comput. complex.

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A characterization of incomplete sequences in vector spaces

Nguyen

2012

Journal of Combinatorial Theory, Series A

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A sequence A of elements an additive group G is incomplete if there exists a group element that can not be expressed as a sum of elements from A. The study of incomplete sequences is a popular topic in combinatorial number theory. However, the structure of incomplete sequences is still far from being understood, even in basic groups.The main goal of this paper is to give a characterization of incomplete sequences in the vector space F d p , where d is a fixed integer and p is a large prime. As an application, we give a new proof for a recent result by Gao-Ruzsa-Thangadurai on the Olson's constant of F 2 p and partially answer their conjecture concerning F 3 p .

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Improved Approximation of Linear Threshold Functions

Diakonikolas

Servedio

2009

2009 24th Annual IEEE Conference on Computational Complexity

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We prove two main results on how arbitrary linear threshold functions f (x) = sign(w · x − θ) over the n-dimensional Boolean hypercube can be approximated by simple threshold functions.Our first result shows that every n-variable threshold function f is ǫ-close to a threshold function depending only on Inf(f ) 2 · poly(1/ǫ) many variables, where Inf(f ) denotes the total influence or average sensitivity of f. This is an exponential sharpening of Friedgut's well-known theorem [Fri98], which states that every Boolean function f is ǫ-close to a function depending only on 2 O(Inf(f )/ǫ) many variables, for the case of threshold functions. We complement this upper bound by showing that Ω(Inf(f ) 2 + 1/ǫ 2 ) many variables are required for ǫ-approximating threshold functions.Our second result is a proof that every n-variable threshold function is ǫ-close to a threshold function with integer weights at most poly(n) · 2Õ (1/ǫ 2/3 ) . This is a significant improvement, in the dependence on the error parameter ǫ, on an earlier result of [Ser07] which gave a poly(n) · 2Õ (1/ǫ 2 ) bound. Our improvement is obtained via a new proof technique that uses strong anti-concentration bounds from probability theory. The new technique also gives a simple and modular proof of the original [Ser07] result, and extends to give low-weight approximators for threshold functions under a range of probability distributions beyond just the uniform distribution.

show abstract

A Structural Approach to Subset-Sum Problems

Cited by 5 publications

References 58 publications

Improved Approximation of Linear Threshold Functions

Improved Approximation of Linear Threshold Functions

A characterization of incomplete sequences in vector spaces

Improved Approximation of Linear Threshold Functions

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