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Playing Unique Games on Certified Small-Set Expanders [article]

Mitali Bafna, Boaz Barak, Pravesh Kothari, Tselil Schramm, David Steurer
2021 arXiv   pre-print
Our algorithm is in fact more versatile, and succeeds even when the constraint graph is not a small-set expander as long as the structure of non-expanding small sets is (informally speaking) "characterized  ...  We give an algorithm for solving unique games (UG) instances whenever low-degree sum-of-squares proofs certify good bounds on the small-set-expansion of the underlying constraint graph via a hypercontractive  ...  In the case of the Johnson graph, which is not a small set expander, we have to work harder.  ... 
arXiv:2006.09969v3 fatcat:d4odahofjjdkdkspwrwy2irswm

On a problem by Shapozenko on Johnson graphs [article]

Víctor Diego, Oriol Serra, Lluís Vena
2017 arXiv   pre-print
Shapozenko asked about the isoperimetric function μ_n,m(k) of Johnson graphs, that is, the cardinality of the smallest boundary of sets with k vertices in J(n,m) for each 1< k<n m.  ...  The Johnson graph J(n,m) has the m--subsets of {1,2,...,n} as vertices and two subsets are adjacent in the graph if they share m-1 elements.  ...  Acknowledgements The authors are grateful to the comments and remarks of the referees, pointing out some inaccuracies in the original manuscript and helping to improve the readability of the paper.  ... 
arXiv:1604.05084v2 fatcat:7ryai42b4nf2fjatc3i6wfurle

A Log-Sobolev Inequality for the Multislice, with Applications

Yuval Filmus, Ryan O'Donnell, Xinyu Wu, Michael Wagner
2018 Innovations in Theoretical Computer Science  
From this, we derive some consequences for small-set expansion and isoperimetry in the multislice, including a KKL Theorem, a Kruskal-Katona Theorem for the multislice, a Friedgut Junta Theorem, and a  ...  We show that the log-Sobolev constant κ for the chain satisfies which is sharp up to constants whenever is constant.  ...  is similar to the random walk in generalized Johnson graphs.  ... 
doi:10.4230/lipics.itcs.2019.34 dblp:conf/innovations/FilmusOW19 fatcat:bkvdwesdobe7nab3ajopcduo4q

Towards a General Direct Product Testing Theorem

Elazar Goldenberg, Karthik C. S., Michael Wagner
2018 Foundations of Software Technology and Theoretical Computer Science  
a member of the Johnson graph family.  ...  Dinur and Kaufman (FOCS '17) analyzed it for the case where V is the set of faces of a Ramanujan complex, where in this case In this paper, we study the testability of direct products in a general setting  ...  We also thank the anonymous reviewers for their detailed and useful feedback.  ... 
doi:10.4230/lipics.fsttcs.2018.11 dblp:conf/fsttcs/GoldenbergS18 fatcat:pxxf25zstnbwzkhqbljcs6q2ru

High Dimensional Expanders: Eigenstripping, Pseudorandomness, and Unique Games [article]

Mitali Bafna, Max Hopkins, Tali Kaufman, Shachar Lovett
2021 arXiv   pre-print
the art [RBS11, ABS15] from nearly-exponential to polynomial time (e.g. for sparsifications of Johnson graphs or of slices of the q-ary hypercube).  ...  Our characterization of expansion also holds an interesting connection to hardness of approximation, where an ℓ_∞-variant for the Grassmann graphs was recently used to resolve the 2-2 Games Conjecture  ...  The Johnson and Grassmann graphs, for instance, are well known to have small non-expanding sets.  ... 
arXiv:2011.04658v3 fatcat:22ksh6av4fdnfpks5empe3f4ri

On a Problem by Shapozenko on Johnson Graphs

Víctor Diego, Oriol Serra, Lluís Vena
2018 Graphs and Combinatorics  
Shapozenko asked about the isoperimetric function µn,m(k) of Johnson graphs, that is, the cardinality of the smallest boundary of sets with k vertices in J(n, m) for each 1 ≤ k ≤ n m .  ...  The Johnson graph J(n, m) has the m-subsets of {1, 2, . . . , n} as vertices and two subsets are adjacent in the graph if they share m−1 elements.  ...  Acknowledgements The authors are grateful to the comments and remarks of the referees, pointing out some inaccuracies in the original manuscript and helping to improve the readability of the paper.  ... 
doi:10.1007/s00373-018-1923-7 fatcat:o762ushckjfmlcvkgbsnnrcmge

Improved Decoding of Expander Codes [article]

Xue Chen, Kuan Cheng, Xin Li, Minghui Ouyang
2022 arXiv   pre-print
Finally, we also give a bound on the list-decoding radius of general expander codes, which beats the classical Johnson bound in certain situations (e.g., when the graph is almost regular and the code has  ...  Our techniques exploit novel combinatorial properties of bipartite expander graphs. In particular, we establish a new size-expansion tradeoff, which may be of independent interests.  ...  We remark that, the Johnson bound r = d/2 + Θ(d 2 /N ) when d is small.  ... 
arXiv:2111.07629v4 fatcat:aqptich7e5dqvaguicthwkvkh4

The covering radius of doubled 2-designs in 2 Ok

Patrick Solé, Arif Ghafoor
1991 Discrete Applied Mathematics  
The upper bound is obtained by generalizing the concept of q-covering in Johnson graphs to the graphs 2ok. We use probabilistic arguments analogous to the Norse bounds of coding theory.  ...  The following problem originated from interconnection network considerations: what is the graphical covering radius of a doubled 2-design in the antipodal double cover of the odd graph 20k?  ...  In the Johnson graph vertices are represented as k-sets chosen from a given u-set, and two vertices are connected if they have k -1 elements in common [4, 9] .  ... 
doi:10.1016/0166-218x(91)90117-f fatcat:7qwg7sovx5ekjfvo7y6jxqgq2m

Towards a General Direct Product Testing Theorem [article]

Elazar Goldenberg, Karthik C. S.
2019 arXiv   pre-print
In this paper, we study the testability of direct products in a general setting, addressing the question: what properties of the domain and the test graph allow one to prove a direct product testing theorem  ...  Note that the above distribution may be viewed as a weighted graph over the vertex set V and is referred to as a test graph.  ...  We also thank the anonymous reviewers for their detailed and useful feedback.  ... 
arXiv:1901.06220v1 fatcat:cyey6msjafckrb34coqkl5bga4

Page 5383 of Mathematical Reviews Vol. , Issue 2003g [page]

2003 Mathematical Reviews  
(UKR-KIEVM-PS; Kiev) An expansion in a small parameter of the probability that a random determinant in the field GF(2) is equal to one. (Ukrainian. Ukrainian summary) Teor. Imovir. Mat. Stat.  ...  An expansion in a small parameter ¢ of the probability of event {A, = 1} is investigated. It is shown that for |x; ;|<T7,, i,j€1, Tyr <0co, n>], P{A, = 1} = P(n) +ef Here P(n) = 4... peek!  ... 

Transitive bounded-degree 2-expanders from regular 2-expanders [article]

Eyal Karni, Tali Kaufman
2020 arXiv   pre-print
In this work, we present a class of bounded degree 2-dimensional expanders, which is the result of a small 2-complex action on a vertex set.  ...  The family of expanders that we get is explicit if the one-skeleton of the small complex is a complete multipartite graph, and it is random in the case of (almost) general d-regular complex.  ...  J(S, n) is the Johnson graph V (J) = S n and v ∼ v ′ if |v ∩ v ′ | = n − 1 For example, in case n = 2 {a, b} ∼ {c, d} if both sets share one element The Johnson graph is a well studied object, and appear  ... 
arXiv:2004.11429v1 fatcat:tu7nocmkufh7hlgnostth2byhu

Common adversaries form alliances: modelling complex networks via anti-transitivity [article]

Anthony Bonato, Ewa Infeld, Hari Pokhrel, Pawel Pralat
2017 arXiv   pre-print
The Iterated Local Anti-Transitivity (or ILAT) model creates anti-clone nodes in each time-step, and joins anti-clones to the parent node's non-neighbor set.  ...  The graphs generated by ILAT exhibit familiar properties of complex networks such as densification, short distances (bounded by absolute constants), and bad spectral expansion.  ...  For instance, the presence of small (3-element) dominating sets suggest the emergence of nodes we describe as su- perpowers, which have broad influence in the network.  ... 
arXiv:1704.05658v1 fatcat:zygza2o37nhxxnlbdjfwuk4tuq

Efficient and Robust Compressed Sensing Using Optimized Expander Graphs

Sina Jafarpour, Weiyu Xu, Babak Hassibi, Robert Calderbank
2009 IEEE Transactions on Information Theory  
In this paper, we improve upon this result by considering expander graphs with expansion coefficient beyond 3 4 and show that, with the same number of measurements, only O(k) recovery iterations are required  ...  We also show that by tolerating a small penalty on the number of measurements, and not on the number of recovery iterations, one can use the efficient construction of a family of expander graphs to come  ...  ACKNOWLEDGMENT The authors would like to thank Piotr Indyk, Justin Romberg, the readers and the anonymous reviewers of the paper for their insights and suggestions.  ... 
doi:10.1109/tit.2009.2025528 fatcat:uafhw7hjqfet3bhb72hpuaaawq

Expanding network communities from representative examples

Andrew Mehler, Steven Skiena
2009 ACM Transactions on Knowledge Discovery from Data  
We present a general method for network community expansion, demonstrating that our methods work well in expanding communities in real world networks starting from small given seed groups (20 to 400 members  ...  Our problem becomes identifying a small conductance subgraph containing many (but not necessarily all) members of the given seed set.  ...  Our evaluator sees the name "Larry Brown" as belonging to a middle infielder that played in the late 1960's, not the current basketball coach.  ... 
doi:10.1145/1514888.1514890 fatcat:n2usjiisrbfofatwutcuudzxqy

Inline function expansion for compiling C programs

P. P. Chang, W.-W. Hwu
1989 Proceedings of the ACM SIGPLAN 1989 Conference on Programming language design and implementation - PLDI '89  
With automatic inline function expansion, programs can be constructed with many small functions to handle complexity and then rely on the compilation to eliminate most of the function calls.  ...  Inline function expansion replaces a function call with the function body.  ...  A weighted call graph G = (N, E, main) is chamcterized by three major components: N is a set of nodes, E is a set of arcs, and main is the first node of the call graph.  ... 
doi:10.1145/73141.74840 dblp:conf/pldi/HwuC89 fatcat:ymh5tm2pybeoto5fewemv5l62y
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