Borel oracles. An analytical approach to constant-time algorithms

Elek, Gábor; Lippner, Gábor

doi:10.1090/s0002-9939-10-10291-3

Cited by 41 publications

(80 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Elek and G. Lippner [8] proved that for a Benjamini-Schramm convergent graph sequence (G i ), the following limit exits:…”

Section: Preliminaries and Basic Notionsmentioning

confidence: 99%

Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems

Csíkvári¹

2017

J. Eur. Math. Soc.

View full text Add to dashboard Cite

Friedland's Lower Matching Conjecture asserts that if G is a d-regular bipartite graph on v(G) = 2n vertices, and m k (G) denotes the number of matchings of size k, thenwhere p = k n . When p = 1, this conjecture reduces to a theorem of Schrijver which says that a d-regular bipartite graph on v(G) = 2n vertices has at leastn perfect matchings. L. Gurvits proved an asymptotic version of the Lower Matching Conjecture, namely he proved that 2010 Mathematics Subject Classification. Primary: 05C35. Secondary: 05C31, 05C70, 05C80.

show abstract

“…Elek and G. Lippner [8] proved that for a Benjamini-Schramm convergent graph sequence (G i ), the following limit exits:…”

Section: Preliminaries and Basic Notionsmentioning

confidence: 99%

Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems

Csíkvári¹

2017

J. Eur. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…To show that the supremum is attained, we use the result of Elek and Lippner [EL2] mentioned earlier to find a sequence of Borel matchings (M n ) with X M n ⊆ X M n+1 and with M n having no augmenting paths of length less than 4n. Then we use the argument in Lyons and Nazarov [LN] to show that M defined by…”

Section: Fix An Infinite Groupmentioning

confidence: 99%

Ultraproducts of measure preserving actions and graph combinatorics

Conley

Kechris

Tucker-Drob

2012

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

show abstract

“…The following variational representation holds for Φ γ . The maximum is attained on u γ (w) = ∂ x Φ γ (w, X γ (w)), where X γ = (X γ (w)) s≤w≤q is the solution to the following SDE, 15) with the initial condition X γ (s) = x.…”

Section: Variational Representations For φ γ and ψ γmentioning

confidence: 99%

Suboptimality of local algorithms for a class of max-cut problems

Chen¹,

Gamarnik²,

Panchenko³

et al. 2019

Ann. Probab.

View full text Add to dashboard Cite

We show that in random K-uniform hypergraphs of constant average degree, for even K ≥ 4, local algorithms defined as factors of i.i.d. can not find nearly maximal cuts when the average degree is sufficiently large. These algorithms have been used frequently to obtain lower bounds for the max-cut problem on random graphs, but it was not known whether they could be successful in finding nearly maximal cuts. This result follows from the fact that the overlap of any two nearly maximal cuts in such hypergraphs does not take values in a certain non-trivial interval -a phenomenon referred to as the overlap gap property -which is proved by comparing diluted models with large average degree with appropriate fully connected spin glass models, and showing the overlap gap property in the latter setting. This paper considers the problem of algorithmically finding nearly optimal spin configurations in the diluted K-spin model. We specifically focus on local algorithms defined as factors of i.i.d., the formal definition of which is provided in Section 2. The diluted K-spin model is also known as the max-cut problem for Kuniform Erdős-Rényi hypergraphs of constant average degree, and also as the random K-XORSAT model. The problem is only interesting for even K and we prove that, for even K ≥ 4, local algorithms fail to find the nearly optimal spin configurations (maximal cuts) once the average degree is large enough.The proof is based on finding a structural constraint for the overlap of any two nearly optimal spin configurations -the overlap gap property -that goes against certain properties of local algorithms. For K = 2, the overlap gap property is not expected to hold, which is why this case is excluded. The structural constraint is derived from recent results on the mean field K-spin spin glass models, in particular, the Parisi formula and the Guerra-Talagrand replica symmetry breaking bound at zero temperature. We begin with a discussion of the model and the notion of algorithms that we use.The K-spin model The set of ±1 spin configurations on N vertices will be denoted by

show abstract

Borel oracles. An analytical approach to constant-time algorithms

Cited by 41 publications

References 12 publications

Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems

Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems

Ultraproducts of measure preserving actions and graph combinatorics

Suboptimality of local algorithms for a class of max-cut problems

Contact Info

Product

Resources

About