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Register a new userLower Bounds for Embedding into Distributions over Excluded Minor Graph Families
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Added by
Ashish Goel
Publication date
2008
fields
Informatics Engineering
and research's language is
English
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It was shown recently by Fakcharoenphol et al that arbitrary finite metrics can be embedded into distributions over tree metrics with distortion O(log n). It is also known that this bound is tight since there are expander graphs which cannot be embedded into distributions over trees with better than Omega(log n) distortion. We show that this same lower bound holds for embeddings into distributions over any minor excluded family. Given a family of graphs F which excludes minor M where |M|=k, we explicitly construct a family of graphs with treewidth-(k+1) which cannot be embedded into a distribution over F with better than Omega(log n) distortion. Thus, while these minor excluded families of graphs are more expressive than trees, they do not provide asymptotically better approximations in general. An important corollary of this is that graphs of treewidth-k cannot be embedded into distributions over graphs of treewidth-(k-3) with distortion less than Omega(log n). We also extend a result of Alon et al by showing that for any k, planar graphs cannot be embedded into distributions over treewidth-k graphs with better than Omega(log n) distortion.
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We study the space complexity of sketching cuts and Laplacian quadratic forms of graphs. We show that any data structure which approximately stores the sizes of all cuts in an undirected graph on $n$ vertices up to a $1+epsilon$ error must use $Omega(nlog n/epsilon^2)$ bits of space in the worst case, improving the $Omega(n/epsilon^2)$ bound of Andoni et al. and matching the best known upper bound achieved by spectral sparsifiers. Our proof is based on a rigidity phenomenon for cut (and spectral) approximation which may be of independent interest: any two $d-$regular graphs which approximate each others cuts significantly better than a random graph approximates the complete graph must overlap in a constant fraction of their edges.
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