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dc.contributor.author
Bandeira, Afonso S.
dc.contributor.author
Banks, Jess
dc.contributor.author
Kunisky, Dmitriy
dc.contributor.author
Moore, Cristopher
dc.contributor.author
Wein, Alexander S.
dc.date.accessioned
2021-01-28T14:37:11Z
dc.date.available
2021-01-26T09:21:49Z
dc.date.available
2021-01-28T14:37:11Z
dc.date.issued
2020-08-27
dc.identifier.uri
http://hdl.handle.net/20.500.11850/465557
dc.description.abstract
We study the problem of efficiently refuting the k-colorability of a graph, or equivalently certifying a lower bound on its chromatic number. We give formal evidence of average-case computational hardness for this problem in sparse random regular graphs, showing optimality of a simple spectral certificate. This evidence takes the form of a computationally-quiet planting: we construct a distribution of d-regular graphs that has significantly smaller chromatic number than a typical regular graph drawn uniformly at random, while providing evidence that these two distributions are indistinguishable by a large class of algorithms. We generalize our results to the more general problem of certifying an upper bound on the maximum k-cut. This quiet planting is achieved by minimizing the effect of the planted structure (e.g. colorings or cuts) on the graph spectrum. Specifically, the planted structure corresponds exactly to eigenvectors of the adjacency matrix. This avoids the pushout effect of random matrix theory, and delays the point at which the planting becomes visible in the spectrum or local statistics. To illustrate this further, we give similar results for a Gaussian analogue of this problem: a quiet version of the spiked model, where we plant an eigenspace rather than adding a generic low-rank perturbation. Our evidence for computational hardness of distinguishing two distributions is based on three different heuristics: stability of belief propagation, the local statistics hierarchy, and the low-degree likelihood ratio. Of independent interest, our results include general-purpose bounds on the low-degree likelihood ratio for multi-spiked matrix models, and an improved low-degree analysis of the stochastic block model.
en_US
dc.language.iso
en
en_US
dc.publisher
Cornell University
en_US
dc.title
Spectral Planting and the Hardness of Refuting Cuts, Colorability, and Communities in Random Graphs
en_US
dc.type
Working Paper
ethz.journal.title
arXiv
ethz.pages.start
2008.12237
en_US
ethz.size
59 p.
en_US
ethz.identifier.arxiv
2008.12237
ethz.publication.place
Ithaca, NY
en_US
ethz.publication.status
published
en_US
ethz.leitzahl
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02502 - Institut für Operations Research / Institute for Operations Research::09679 - Bandeira, Afonso / Bandeira, Afonso
en_US
ethz.leitzahl.certified
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02502 - Institut für Operations Research / Institute for Operations Research::09679 - Bandeira, Afonso / Bandeira, Afonso
en_US
ethz.relation.isPreviousVersionOf
handle/20.500.11850/527834
ethz.date.deposited
2021-01-26T09:21:56Z
ethz.source
FORM
ethz.eth
yes
en_US
ethz.availability
Metadata only
en_US
ethz.rosetta.installDate
2021-01-28T14:37:25Z
ethz.rosetta.lastUpdated
2024-02-02T12:59:39Z
ethz.rosetta.versionExported
true
ethz.COinS
ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.atitle=Spectral%20Planting%20and%20the%20Hardness%20of%20Refuting%20Cuts,%20Colorability,%20and%20Communities%20in%20Random%20Graphs&rft.jtitle=arXiv&rft.date=2020-08-27&rft.spage=2008.12237&rft.au=Bandeira,%20Afonso%20S.&Banks,%20Jess&Kunisky,%20Dmitriy&Moore,%20Cristopher&Wein,%20Alexander%20S.&rft.genre=preprint&
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