Given a weighted graph $G=(V,E,w)$, a partition of $V$ is $\Delta$-bounded if the diameter of each cluster is bounded by $\Delta$. A distribution over $\Delta$-bounded partitions is a $\beta$-padded decomposition if every ball of radius $\gamma\Delta$ is contained in a single cluster with probability at least $e^{-\beta\cdot\gamma}$. 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 $\left(O(d),\tilde{O}(d)\right)$-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.
翻译:根据一个加权的GG=(V,E,w)美元,如果每组的直径被$\Delta美元捆绑,那么,折成的美元是美元。如果每组的直径被$\Delta美元绑定,那么,每组的软直径是美元=Delta美元。如果每块半径$\Delta=(V,E,w)美元(美元=G,美元=Delta美元)的分布是美元=Betggggg =(美元=(V,E,E,E,wdo)美元=美元=(美元=Delta美元),那么,如果每块半径直径直径为$=Delta 美元。如果每块半径的每颗粒子(美元=G,美元=Delta美元=G)的分布是美元-ptregtal =$(美元)的微直径直径直径直径测量值,那么对于硬直径直径的直径直径直径直径直径直径直径直为$(r=G美元)。