The asymptotic behaviour of Lovász' theta-function for random graphs.

In the asymptotic case, shared randomness is shown to be just as powerful as arbitrary non-signalling correlations for noisy channel simulation, which is not true for the asymptotic zero-error capacities ... The most striking result of this kind is that entanglement can assist in zero-error communication, in stark contrast to the standard setting of communicaton with asymptotically vanishing error in which ... ACKNOWLEDGMENTS We would like to thank Nicolas Brunner, Runyao Duan, Tsuyoshi Ito, Ashley Montanaro, Marcin Pawłowski, Paul Skrzypczyk and Stephanie Wehner for useful discussions. ...

doi:10.1109/tit.2011.2159047 fatcat:w3um3pdz3fal3d25il22gjlj5m

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As a result of this study, connections between the Lov\'asz theta function, the expurgated bound of Gallager, the cutoff rate of a classical channel and the sphere packing bound for classical-quantum channels ... The contribution of the paper goes in two main directions: i) extending classical bounds of Shannon, Gallager and Berlekamp to classical-quantum channels, and ii) proposing a new framework for lower bounding ... There, the authors extend the algebraic definition of the Lovász theta function to consider what they call non-commutative graphs. ...

doi:10.1109/tit.2013.2283794 fatcat:vcveintqurhwzmw4nbw3kosjhu

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In practice, we report encouraging computational results on random graphs, Knesser graphs, "forbidden intersection" graphs, the Toronto benchmark, and the International Timetabling Competition. ... We present SDP relaxations for a variety of mutual-exclusion scheduling and timetabling problems, starting from a bound on the number of tasks executed within each period, which corresponds to graph colouring ... Random Graphs Next, we show that the same behaviour can be observed on a large sample of random graphs. ...

arXiv:1904.03539v1 fatcat:csewztv5draqhp7zdx5foqtsbq

the Lovász number and the fractional Haemers bounds are in fact elements of the asymptotic spectrum (spectral points). ... He used it to obtain a dual characterization of the Shannon capacity of graphs as the minimum of the evaluation function over the asymptotic spectrum and noted that several known upper bounds, including ... The probabilistic refinement of the Lovász theta number was defined and studied by Marton in [Mar93] via a non-asymptotic formula. ...

arXiv:1903.01857v1 fatcat:2jvbzuz5zfa7hdkst3fotwrmhq

We show that for the standard setting of the parameters of the problem, namely, a clique of size k = √(n) planted in a random G(n, 1/2) graph, the known polynomial time algorithms can be extended (in a ... In the well known planted clique problem, a clique (or alternatively, an independent set) of size k is planted at random in an Erdos-Renyi random G(n, p) graph, and the goal is to design an algorithm that ... Acknowledgements The work of Uriel Feige is supported in part by the Israel Science Foundation (grant No. 1388/16). ...

arXiv:2004.12258v2 fatcat:7b6htco4wfaolauf5rjg3xoky4

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We investigate the asymptotic version of the Erd\H{o}s-Ko-Rado theorem for the random $k$-uniform hypergraph $\mathcal{H}^k(n,p)$. For $2 \leq k(n) \leq n/2$, let $N=\binom{n}k$ and $D=\binom{n-k}k$. ... This lower bound on $p$ is asymptotically best possible for $k=\Theta(n)$. For this range of $k$ and $p$, we are able to show stability as well. A different behavior occurs when $k = o(n)$. ... Many thanks go to Oded Regev and Ehud Friedgut for sharing a preliminary version of their paper. ...

arXiv:1409.3634v2 fatcat:rpxzwpso2rez5guufrryzjp4hy

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We thank a referee for his careful reading and his suggestions that helped improve the presentation of this chapter. ... Moreover, the integrality ratio is asymptotically equal to 1 for the random graphs G n,p (p denoting the edge probability) [64] . ... The theta function ϑ(G) and the basic semidefinite relaxation TH(G) Lovász [164] introduced the following parameter ϑ(G), known as the theta number: ϑ(G) := max e T Xe s.t. ...

doi:10.1016/s0927-0507(05)12008-8 fatcat:ez23hvr5znfolnppcugzpevgpu

One of the main reasons for this growth is the tight connection between Discrete Mathematics and Theoretical Computer Science, and the rapid development of the latter. ... This is a survey of two of the main general techniques that played a crucial role in the development of modern combinatorics; algebraic methods and probabilistic methods. ... The asymptotic behavior of these numbers for graphs of chromatic number at least 3 is well known, see, e.g., [15] . ...

arXiv:math/0212390v1 fatcat:v4iw43rdzbgjroxo37nsl45nle

A different connection between kernels and random graphs arises in the recent work of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi. ... Roughly speaking, there is one model for each kernel, i.e., each symmetric measurable function from [0,1]^2 to the non-negative reals, although the details are much more complicated. ... The authors would like to thank an anonymous referee for many detailed suggestions improving the presentation of the paper. ...

arXiv:0708.1919v3 fatcat:6mead2polvbohd4hbv5z7kzdhq

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In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. ... The number and importance of these results is so fascinating that it makes sense to present this survey. ... Juhász, who also analysed the behaviour of the theta-function on random graphs, and introduced the eigenvalues in the clustering. ...

doi:10.1007/978-1-4613-8354-3_5 fatcat:btv4vlhelnd3tfjebil7nktmzq

that they have a colouring of the graph. ... We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. ... Acknowledgments The authors thank Harry Buhrman, Matthias Christandl, Sean Clark, Patrick Hayden and Troy Lee for dis-cussions on various aspects of this paper; in particular thanks to Ronald de Wolf for ...

arXiv:quant-ph/0608016v3 fatcat:oturm32pe5ekzfn6e2avwzaadm

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Lovász and Schrijver [21] introduced the systems LS and LS + for systematically tightening LP and SDP relaxations, respectively, over many rounds. ... In particular, we prove an integrality gap of 2 − o(1) for VERTEX COVER SDPs obtained by tightening the standard LP relaxation with Ω( log n/ log log n) rounds of LS + . ... with [19] studying the Lovász theta function). ...

doi:10.1137/080721479 fatcat:mlwlm5rrsfairhxl6zb5icmg2e

Hence, our communication protocol is similar to the Holevo-Schumacher-Westmoreland protocol. We reformulate the problem of finding the QZEC in terms of graph theory. ... In particular, we exhibit a quantum channel for which we claim the QZEC can only be reached by a set of non-orthogonal states. ... Lovász theta function The redefinition of the zero-error capacity in terms of graph has yielded interesting constructions in combinatorics and graph theory. ...

doi:10.1007/978-3-319-42794-2_5 fatcat:44z4df4l4fci7pipjfqiqctkw4

The k^th power of a graph G=(V,E), G^k, is the graph whose vertex set is V and in which two distinct vertices are adjacent if and only if their distance in G is at most k. ... Some of the bounds previously known in the literature follow as a corollary of our main results. Infinite families of graphs where the bounds are sharp are presented as well. ... The authors would also like to thank Anurag Bishnoi for noticing a tight family for our bound (19) . ...

arXiv:2010.12649v1 fatcat:lhgbg3vhwbgkjc73cox7szpbre

One implication of the result is a (1/2)-factor approximation gap, asymptotically in d, for estimating the density of maximal induced forests in locally tree-like d-regular graphs via factor of iid processes ... We show that the density of any factor of iid site percolation process with finite clusters is asymptotically at most (log d)/d as d → ∞. ... The author is grateful to Endre Csóka and Viktor Harangi for explaining the construction of the optimal percolation process of density 3/4 in Section 4.5. ...

doi:10.1137/15m1021362 fatcat:pfxadnn555ggrclkyrnghcfic4

Zero-Error Channel Capacity and Simulation Assisted by Non-Local Correlations

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Lower Bounds on the Probability of Error for Classical and Classical-Quantum Channels

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Semidefinite Programming in Timetabling and Mutual-Exclusion Scheduling [article]

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Probabilistic refinement of the asymptotic spectrum of graphs [article]

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How to hide a clique? [article]

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Erdős-Ko-Rado for random hypergraphs: asymptotics and stability [article]

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Semidefinite Programming and Integer Programming [chapter]

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Discrete mathematics: methods and challenges [article]

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Metrics for sparse graphs [article]

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Eigenvalues in Combinatorial Optimization [chapter]

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On the quantum chromatic number of a graph [article]

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Integrality Gaps of $2-o(1)$ for Vertex Cover SDPs in the Lovász–Schrijver Hierarchy

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Zero-Error Capacity of Quantum Channels [chapter]

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Optimization of eigenvalue bounds for the independence and chromatic number of graph powers [article]

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Factor of IID Percolation on Trees

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