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Gap Amplification for Small-Set Expansion via Random Walks
[article]

2014
*
arXiv
*
pre-print

In this work, we achieve

arXiv:1310.1493v3
fatcat:7c6rayjlhrfbroaev63vr67hti
*gap**amplification**for*the*Small*-*Set**Expansion*problem. ... We achieve this*amplification**via**random**walks*-- our gadget is the graph with adjacency matrix corresponding to a*random**walk*on the original graph. ... In this work, we show that*random**walks*can be used to achieve*gap**amplification**for**small**set**expansion*. ...##
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Gap Amplification for Small-Set Expansion via Random Walks *

unpublished

In this work, we achieve

fatcat:tkpz5xpjnjcjjofgsr54xvd4xi
*gap**amplification**for*the*Small*-*Set**Expansion*problem. ... We achieve this*amplification**via**random**walks*-the output graph corresponds to taking*random**walks*on the original graph. ... In this work, we show that*random**walks*can be used to achieve*gap**amplification**for**small**set**expansion*. ...##
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The PCP Theorem for NP Over the Reals

2014
*
Foundations of Computational Mathematics
*

The analogue result holds

doi:10.1007/s10208-014-9188-x
fatcat:v3hhmowbfnhm3lnqsp7ff6aaea
*for*the complex numbers and NP C . ... Our proof structurally follows the one by Dinur*for*classical NP. However, a lot of minor and major changes are necessary due to the real numbers as underlying computational structure. ...*For*the definition of algebraic*expansion*we need the*random**walk*matrix of a graph G. ...##
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Reductions between Expansion Problems

2012
*
2012 IEEE 27th Conference on Computational Complexity
*

We also show a "hardness

doi:10.1109/ccc.2012.43
dblp:conf/coco/RaghavendraST12
fatcat:7udhq2bz2fevzpafyvlphlnpum
*amplification*" result*for**Small*-*Set**Expansion*proving that if the*Small*-*Set**Expansion*Hypothesis holds then the current best algorithm*for**Small*-*Set**Expansion*due to [8] is optimal ...*For*all η > 0, M 1 and all δ < 1/M, there is polynomial time reduction from*Small*-*Set**Expansion*( η M , δ) to*Small*-*Set**Expansion*(η, δ, M). III. Technical Preliminaries*Random**walks*on graphs. ...##
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The PCP theorem by gap amplification

2006
*
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing - STOC '06
*

Our main theorem is the aforementioned '

doi:10.1145/1132516.1132553
dblp:conf/stoc/Dinur06
fatcat:sfuuzyozobdhvd3off4pwn7cq4
*gap**amplification*step', where a graph G is converted into a new graph G whose unsat value is doubled. ... The*amplification*step causes an increase in alphabet-size that is corrected by a (standard) PCP composition step. Iterative application of these two steps yields a proof*for*the PCP theorem. ... Acknowledgements I am thankful to Omer Reingold and Luca Trevisan*for*many discussions, especially ones about combinatorial analyses of graph powering, which were the direct trigger*for*the*amplification*...##
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The PCP theorem by gap amplification

2007
*
Journal of the ACM
*

Our main theorem is the aforementioned '

doi:10.1145/1236457.1236459
fatcat:l2wnhmpjwfaazopk37x4prkq6m
*gap**amplification*step', where a graph G is converted into a new graph G whose unsat value is doubled. ... The*amplification*step causes an increase in alphabet-size that is corrected by a (standard) PCP composition step. Iterative application of these two steps yields a proof*for*the PCP theorem. ... Acknowledgements I am thankful to Omer Reingold and Luca Trevisan*for*many discussions, especially ones about combinatorial analyses of graph powering, which were the direct trigger*for*the*amplification*...##
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Reductions Between Expansion Problems
[article]

2010
*
arXiv
*
pre-print

The key technical ingredient is a new way of exploiting the structure of the Unique Games instances obtained from the

arXiv:1011.2586v1
fatcat:j2hrhg55ufebpija3lpdv7r76m
*Small*-*Set**Expansion*Hypothesis*via*(Raghavendra, Steurer, 2010). ... The*Small*-*Set**Expansion*Hypothesis (Raghavendra, Steurer, STOC 2010) is a natural hardness assumption concerning the problem of approximating the edge*expansion*of*small**sets*in graphs. ... Acknowledgments We are grateful to Subhash Khot*for*suggesting that our techniques should also show that Unique Games is SSE-hard on graph with high (*small*-*set*)*expansion*(Theorem 3.2). ...##
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Near-Optimal Cayley Expanders for Abelian Groups
[article]

2021
*
arXiv
*
pre-print

Our technique is an extension of the bias

arXiv:2105.01149v1
fatcat:o3qjth3rbvbt5po3eg4bvoo63y
*amplification*technique of Ta-Shma (2017), who used*random**walks*on expanders to obtain expanding generating*sets*over the additive group of n-bit strings. ...*gap*. ... Like Ta-Shma, their technique is to use bias*amplification**via*expander graphs; specifically, they amplify bias*via*an iterated application of a 1-step*random**walk*on an expander graph. ...##
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Chernoff Bound for High-Dimensional Expanders

2020
*
International Workshop on Approximation Algorithms for Combinatorial Optimization
*

Because of these obstructions, the spectral

doi:10.4230/lipics.approx/random.2020.25
dblp:conf/approx/KaufmanS20
fatcat:rb3g5doekzf75nimvxk6bakd5y
*gap*of high-order*random**walks*is inherently*small*. ... Given a graph G and a function f on the vertices, it states that the probability of f's mean sampled*via*a*random**walk*on G to deviate from its actual mean, has a bound that depends on the spectral*gap*... Because of these obstructions, the spectral*gap*1 − λ(M ) of high-order*random**walks*is inherently*small*. In this paper, we manage to overcome this problem by looking beyond the spectral*gap*. ...##
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StoqMA vs. MA: the power of error reduction
[article]

2021
*
arXiv
*
pre-print

Although error reduction is commonplace

arXiv:2010.02835v3
fatcat:xe7grx2ymbepvcsih5d7hijjja
*for*many complexity classes, such as BPP, BQP, MA, QMA, etc.,this property remains open*for*StoqMA since Bravyi, Bessen and Terhal defined this class in 2006. ... Part of this work was done while A.G. was affiliated to CWI and QuSoft and part of it was done while A.G. was visiting the Simons Institute*for*the Theory of Computing. ... we leave this*set*is negligibly*small*. ...##
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Expander graphs and their applications

2006
*
Bulletin of the American Mathematical Society
*

From the probabilistic viewpoint, one considers the natural

doi:10.1090/s0273-0979-06-01126-8
fatcat:5u65bvautndzhnkhxurqrudqfy
*random**walk*on a graph, in which we have a token on a vertex, that moves at every step to a*random*neighboring vertex, chosen uniformly and independently ... Equivalently, using the geometric notion of isoperimetry, every*set*of vertices has a (relatively) very large boundary. ... Avi Wigderson is a professor at the Institute*for*Advanced Study in Princeton. Before that he was a professor of computer science at the Hebrew University*for*fifteen years. ...##
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Stoquastic PCP vs. Randomness
[article]

2019
*
arXiv
*
pre-print

Thus, if there exists a

arXiv:1901.05270v2
fatcat:qcryivy4sjcgpe6w5j4gxjngta
*gap*-*amplification*procedure*for*uniform stoquastic Local Hamiltonians (in analogy to the*gap**amplification*procedure*for*constraint satisfaction problems in the original PCP theorem ... We feel this work opens up a rich*set*of new directions to explore, which might lead to progress on both quantum PCP and derandomization. We also provide two*small*side results of potential interest. ... The*random**walk*proceeds by repeating this process with x 1 . We describe the*random**walk*proposed by BT (simplified*for*the uniform case) in Figure 3 . 1. Let x 0 be the initial string. ...##
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The unified theory of pseudorandomness

2007
*
ACM SIGACT News
*

Pseudorandomness is the theory of efficiently generating objects that "look

doi:10.1145/1324215.1324225
fatcat:yluo6pb23jd53kyyp2yofuv6iy
*random*" despite being constructed with little or no*randomness*. ... Nevertheless, depending on the parameters, vertex*expansion*(as in Definition 6 and Proposition 7) often implies stronger measures of*expansion*(such as a spectral*gap*[Alo] and*randomness*condensing ... vertices, with A > 1 and*expansion*achieved*for**sets*of size up to K = Ω(M ). ...##
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Two combinatorial MA-complete problems
[article]

2021
*
arXiv
*
pre-print

We notice that the main result of Aharonov and Grilo carries over to the SetCSP problem in a straightforward way, implying that finding a

arXiv:2003.13065v3
fatcat:qoxrkm2uxjf4rmvjqlhc2jxg6q
*gap*-*amplification*procedure*for*SetCSP (as in Dinur's PCP proof ... The fact that the first, more natural, problem of ACAC is MA-hard follows quite naturally from this proof, while the containment of ACAC in MA is based on the theory of*random**walks*. ... We are grateful to Umesh Vazirani D.A. is grateful*for*the support of ISF grant 1721/17. Part of this work was done while A.G. was visiting the Simons Institute*for*the Theory of Computing. ...##
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The Unified Theory of Pseudorandomness

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

Pseudorandomness is the theory of efficiently generating objects that "look

doi:10.1142/9789814324359_0165
fatcat:2c3r7mxikfbttbghccecdufd3u
*random*" despite being constructed with little or no*randomness*. ... Nevertheless, depending on the parameters, vertex*expansion*(as in Definition 6 and Proposition 7) often implies stronger measures of*expansion*(such as a spectral*gap*[Alo] and*randomness*condensing ... vertices, with A > 1 and*expansion*achieved*for**sets*of size up to K = Ω(M ). ...
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