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

Abstract

Given a metric space $(X,d_X)$, a $(β,s,Δ)$-sparse cover is a collection of clusters $\mathcal{C}\subseteq P(X)$ with diameter at most $Δ$, such that for every point $x\in X$, the ball $B_X(x,\fracΔβ)$ is fully contained in some cluster $C\in \mathcal{C}$, and $x$ belongs to at most $s$ clusters in $\mathcal{C}$. Our main contribution is to show that the shortest path metric of every $K_r$-minor free graphs admits $(O(r),O(r^2),Δ)$-sparse cover, and for every $ε>0$, $(4+ε,O(\frac1ε)^r,Δ)$-sparse cover (for arbitrary $Δ>0$). We then use this sparse cover to show that every $K_r$-minor free graph embeds into $\ell_\infty^{\tilde{O}(\frac1ε)^{r+1}\cdot\log n}$ with distortion $3+ε$ (resp. into $\ell_\infty^{\tilde{O}(r^2)\cdot\log n}$ 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).

Figures (2)

• Figure 1: (a) Illustration of an $(\Delta, \gamma, w)$-buffered cop decomposition of the unweighted grid graph together with (b) - the associated tree $\mathcal{T}$. There are $9$ different supernodes $\eta_1,\dots,\eta_9$, all colored with different colors. Each supernode $\eta$ contains a shortest path tree $T_\eta$ (the bold lines) with at most $3$ leaves, where all the vertices $x\in\eta$ in the super node are at distance at most $\Delta=6$ from $T_\eta$. The domain of each supernode consist of all the supernodes in its subtree. For example $\mathrm{dom}(\eta_5)=G\left[\eta_5\cup\eta_7\cup\eta_8\cup\eta_9\right]$, and $\mathrm{dom}(\eta_3)=G\left[V\setminus(\eta_1\cup\eta_2)\right]$. As $\eta_3$ and $\eta_9$ are not adjacent, the distance from $\eta_3$ to any vertex in $\eta_9$ (w.r.t. $\mathrm{dom}(\eta_3)$) is at least $\gamma$. The associated digraph $\overrightarrow{G}_{\mathcal{C}}$ is illustrated in (c).
• Figure 2: Illustration of the contraction and expansion proof in \ref{['lem:CURWremoveAspectRatio']}.

Theorems & Definitions (48)

• Definition 1: Sparse Cover
• Theorem 1: Cover for Minor Free Graphs
• Corollary 1
• Theorem 2
• Corollary 2
• Definition 2
• Theorem 3: Buffered cop decompsition
• Definition 3: Transitive DAG
• Lemma 1
• proof