On Sparse Covers of Minor Free Graphs, Low Dimensional Metric Embeddings, and other applications.

Given a metric space (X,dX), a (β,s,Δ)-sparse cover is a collection of clusters C⊆P(X) with diameter at most Δ, such that for every point x∈X, the ball BX(x,Δβ) is fully contained in some cluster C∈C, and x belongs to at most s clusters in C. Our main contribution is to show that the shortest path metric of every Kr-minor free graphs admits (O(r),O(r2),Δ)-sparse cover, and for every ϵ>0, (4+ϵ,O(1ϵ)r,Δ)-sparse cover (for arbitrary Δ>0). We then use this sparse cover to show that every Kr-minor free graph embeds into ℓO~(1ϵ)r+1⋅logn∞ with distortion $3+\eps$ (resp. into ℓO~(r2)⋅logn∞ with distortion O(r)). Further, we provide applications of these sparse covers into padded decompositions, sparse partitions, universal TSP / Steiner tree, oblivious buy at bulk, name independent routing, and path reporting distance oracles.