On Strong Diameter Padded Decompositions

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 β-padded decomposition if every ball of radius γDelta is contained in a single cluster with probability at least e-β·γ. 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 Kr minor free graphs admit weak decompositions with padding parameter O(r), while for strong decompositions only O(r2) padding parameter was known. Furthermore, for the case of a graph G, for which the induced shortest path metric dG 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 Kr 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),tildeO(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.