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

Prasad Raghavendra, Tselil Schramm
2014 arXiv   pre-print
In this work, we achieve 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.  ... 
arXiv:1310.1493v3 fatcat:7c6rayjlhrfbroaev63vr67hti

Gap Amplification for Small-Set Expansion via Random Walks *

Prasad Raghavendra, Tselil Schramm
unpublished
In this work, we achieve 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.  ... 
fatcat:tkpz5xpjnjcjjofgsr54xvd4xi

The PCP Theorem for NP Over the Reals

Martijn Baartse, Klaus Meer
2014 Foundations of Computational Mathematics  
The analogue result holds 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.  ... 
doi:10.1007/s10208-014-9188-x fatcat:v3hhmowbfnhm3lnqsp7ff6aaea

Reductions between Expansion Problems

Prasad Raghavendra, David Steurer, Madhur Tulsiani
2012 2012 IEEE 27th Conference on Computational Complexity  
We also show a "hardness 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.  ... 
doi:10.1109/ccc.2012.43 dblp:conf/coco/RaghavendraST12 fatcat:7udhq2bz2fevzpafyvlphlnpum

The PCP theorem by gap amplification

Irit Dinur
2006 Proceedings of the thirty-eighth annual ACM symposium on Theory of computing - STOC '06  
Our main theorem is the aforementioned '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  ... 
doi:10.1145/1132516.1132553 dblp:conf/stoc/Dinur06 fatcat:sfuuzyozobdhvd3off4pwn7cq4

The PCP theorem by gap amplification

Irit Dinur
2007 Journal of the ACM  
Our main theorem is the aforementioned '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  ... 
doi:10.1145/1236457.1236459 fatcat:l2wnhmpjwfaazopk37x4prkq6m

Reductions Between Expansion Problems [article]

Prasad Raghavendra and David Steurer and Madhur Tulsiani
2010 arXiv   pre-print
The key technical ingredient is a new way of exploiting the structure of the Unique Games instances obtained from the 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).  ... 
arXiv:1011.2586v1 fatcat:j2hrhg55ufebpija3lpdv7r76m

Near-Optimal Cayley Expanders for Abelian Groups [article]

Akhil Jalan, Dana Moshkovitz
2021 arXiv   pre-print
Our technique is an extension of the bias 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.  ... 
arXiv:2105.01149v1 fatcat:o3qjth3rbvbt5po3eg4bvoo63y

Chernoff Bound for High-Dimensional Expanders

Tali Kaufman, Ella Sharakanski, Raghu Meka, Jarosław Byrka
2020 International Workshop on Approximation Algorithms for Combinatorial Optimization  
Because of these obstructions, the spectral 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.  ... 
doi:10.4230/lipics.approx/random.2020.25 dblp:conf/approx/KaufmanS20 fatcat:rb3g5doekzf75nimvxk6bakd5y

StoqMA vs. MA: the power of error reduction [article]

Dorit Aharonov, Alex B. Grilo, Yupan Liu
2021 arXiv   pre-print
Although error reduction is commonplace 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.  ... 
arXiv:2010.02835v3 fatcat:xe7grx2ymbepvcsih5d7hijjja

Expander graphs and their applications

Shlomo Hoory, Nathan Linial, Avi Wigderson
2006 Bulletin of the American Mathematical Society  
From the probabilistic viewpoint, one considers the natural 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.  ... 
doi:10.1090/s0273-0979-06-01126-8 fatcat:5u65bvautndzhnkhxurqrudqfy

Stoquastic PCP vs. Randomness [article]

Dorit Aharonov, Alex B. Grilo
2019 arXiv   pre-print
Thus, if there exists a 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.  ... 
arXiv:1901.05270v2 fatcat:qcryivy4sjcgpe6w5j4gxjngta

The unified theory of pseudorandomness

S. Vadhan
2007 ACM SIGACT News  
Pseudorandomness is the theory of efficiently generating objects that "look 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 ).  ... 
doi:10.1145/1324215.1324225 fatcat:yluo6pb23jd53kyyp2yofuv6iy

Two combinatorial MA-complete problems [article]

Dorit Aharonov, Alex B. Grilo
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 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.  ... 
arXiv:2003.13065v3 fatcat:qoxrkm2uxjf4rmvjqlhc2jxg6q

The Unified Theory of Pseudorandomness

Salil Vadhan
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
Pseudorandomness is the theory of efficiently generating objects that "look 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 ).  ... 
doi:10.1142/9789814324359_0165 fatcat:2c3r7mxikfbttbghccecdufd3u
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