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, Δ/β) 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 ∈ > 0, (4 + ∈,O(1/∈)r, Δ)-sparse cover (for arbitrary Δ > 0). We then use this sparse cover to show that every Kr-minor free graph embeds into ℓÕ (1∈)r+1·log n∞ with distortion 3 + ∈ (resp. into ℓÕ (r2)·log n∞ with distortion O(r)). Further, among other applications, this sparse cover immediately implies an algorithm for the oblivious buy-at-bulk problem in fixed minor free graphs with the tight approximation factor O(log n) (previously nothing beyond general graphs was known).