On Strong Diameter Padded Decompositions

TL;DR

This paper advances the theory of strong diameter padded decompositions by proving that $K_r$-minor-free graphs admit strong $O(r)$-padded decompositions and graphs with doubling dimension ${\rm ddim}$ admit strong $O({\rm ddim})$-padded decompositions, matching the state of the art for weak decompositions in these families. The authors introduce a main technical tool that converts a center-set structure with bounded local overlap into a strong padded decomposition using ball carving with truncated/exponential starting times and a Lovász Local Lemma-based sparsification, enabling efficient construction. They extend these decompositions to tight sparse cover schemes and apply them to key problems, including improving Unique Games approximations on minor-free graphs, constructing light and sparse spanners for doubling metrics, and building path-reporting distance oracles with favorable stretch and query performance. The results significantly improve parameter regimes for strong decompositions, provide tight bounds in doubling metrics, and yield practical algorithmic consequences for a range of distance-inspired graph problems.

Abstract

Given a weighted graph $G=(V,E,w)$, a partition of $V$ is $Δ$-bounded if the diameter of each cluster is bounded by $Δ$. A distribution over $Δ$-bounded partitions is a $β$-padded decomposition if every ball of radius $γΔ$ is contained in a single cluster with probability at least $e^{-β\cdotγ}$. The weak diameter of a cluster $C$ is measured w.r.t. distances in $G$, while the strong diameter is measured w.r.t. distances in the induced graph $G[C]$. The decomposition is weak/strong according to the diameter guarantee. Formerly, it was proven that $K_r$ minor free graphs admit weak decompositions with padding parameter $O(r)$, while for strong decompositions only $O(r^2)$ padding parameter was known. Furthermore, for the case of a graph $G$, for which the induced shortest path metric $d_G$ has doubling dimension $d$, a weak $O(d)$-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known. We construct strong $O(r)$-padded decompositions for $K_r$ minor free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension $d$ we construct a strong $O(d)$-padded decomposition, which is also tight. We use this decomposition to construct strong $\left(O(d),\tilde{O}(d)\right)$ sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles.

Figures (4)

• Figure 1: The figure illustrates the $6$ first steps in \ref{['alg:coreClustering']}. Here $G$ is the (weighted) grid graph. Note that $G$ excludes $K_5$ as a minor. In step $(4)$, $G_4$ is the graph induced by all the vertices not colored in blue, orange or red. $G_4$ has a single connected component $C_4$. The green vertex defined as $x_4$. ${\cal K}_{|C_i}$ consist of $3$ clusters $S_1,S_2,S_3$ colored respectively by blue, orange and red. $T_4$ is a tree rooted in $x_4$ colored in bold green, that consist of 3 shortest paths. Each of $S_1,S_2,S_3$ has a vertex of $T_4$ as a neighbor. $R_4$ is chosen according to $\exp(1)$. The new cluster $S_4$, colored in green, consist of all vertices in $C_4$ at distance at most $R_4\Delta$ from $T_4$ w.r.t. $d_{G_4}$.
• Figure 2: In both figures illustrated an unweighted graph, the clusters $S_1,S_2,S_3$ are colored in blue, orange and cyan respectively. At the forth step a new cluster is created with core $T_4$. On the left, $z$ is a vertex at distance greater than $\alpha\cdot\Delta$ from $T_4$. Hence $\tilde{R}_4$ is defined to be $\tilde{R}_i=\frac{d_{G_4}(z,T_4)}{\Delta}-\alpha$. If $R_i$ is greater than $\tilde{R}_i$, then $T_i$ will join $\mathcal{J}_z$, and we will set $h=\alpha$. In the figure $R_4$ equals $\tilde{R}_4$. Next, the algorithm creates a cluster $S_5$ with core $T_5$, where $d_{G_5}(z,T_5)\le \alpha\cdot \Delta$. Hence $\tilde{R}_5$ is set to be $0$, $T_5$ joins $\mathcal{J}_z$, and $h$ set to be $\frac{1}{\Delta}\cdot d_{G_{5}}(z,T_{5})$. On the right, just before the creation of $S_4$ the vector $\boldsymbol{x}=(x_1,x_2,x_3)$ where $x_1=34$, $x_2=6$, and $x_3=3$ consist of the distances from $z$ to the clusters $S_1,S_2,S_3$ respectively. The potential at this point is $\Phi(\boldsymbol{x})=e^{-3}+e^{-6}+e^{-34}$. After the creation of $S_4$ with $y=3$, there is no longer a path from $S_1$ to $z$, the distance from $S_2$ to $z$ increased from $6$ to $8$, and the distance from $S_3$ to $z$ remained unchanged. The new vector is $\boldsymbol{x}'=(8,3,3)$, and the potential is $\Phi(\boldsymbol{x}')=e^{-3}+e^{-3}+e^{-8}$.
• Figure :
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Theorems & Definitions (41)

• Lemma 1: Packing Property
• Definition 1: Padded Decomposition
• proof
• proof
• Definition 2: Sparse Cover
• Theorem 1
• Claim 1
• Claim 2
• proof
• Claim 3