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A $(β,δ,Δ)$-padded decomposition of an edge-weighted graph $G = (V,E,w)$ is a stochastic decomposition into clusters of diameter at most $$Δ$$ such that for every vertex $$v\in V$$, the probability that $$\rm{ball}_G(v,γΔ)$$ is entirely contained in the cluster containing $$v$$ is at least $$e^{-βγ}$$ for every $$γ\in [0,δ]$$. Padded decompositions have been studied for decades and have found numerous applications, including metric embedding, multicommodity flow-cut gap, multicut, and zero extension problems, to name a few. In these applications, parameter $$β$$, called the padding parameter, is the most important parameter since it decides either the distortion or the approximation ratios. For general graphs with $$n$$ vertices, $$β= Θ(\log n)$$. Klein, Plotkin, and Rao showed that $$K_r$$-minor-free graphs have padding parameter $β= O(r^3)$, which is a significant improvement over general graphs when $$r$$ is a constant. A long-standing conjecture is to construct a padded decomposition for $$K_r$$-minor-free graphs with padding parameter $$β= O(\log r)$$. Despite decades of research, the best-known result is $β= O(r)$, even for graphs with treewidth at most $$r$$. In this work, we make significant progress toward the aforementioned conjecture by showing that graphs with treewidth $$\rm{tw}$$ admit a padded decomposition with padding parameter $$O(\log \rm{tw})$$, which is tight. As corollaries, we obtain an exponential improvement in dependency on treewidth in a host of algorithmic applications: $$O(\sqrt{ \log n \cdot \log(\rm{tw})})$$ flow-cut gap, max flow-min multicut ratio of $$O(\log(\rm{tw}))$$, an $$O(\log(\rm{tw}))$$ approximation for the 0-extension problem, an $$\ell^{O(\log n)}_\infty$$ embedding with distortion $$O(\log \rm{tw})$$, and an $$O(\log \rm{tw})$$ bound for integrality gap for the uniform sparsest cut. 39 pages. This is the TheoretiCS journal version 

Finding the diameter of a graph in general cannot be done in truly subquadratic assuming the Strong Exponential Time Hypothesis (SETH), even when the underlying graph is unweighted and sparse. When restricting to concrete classes of graphs and assuming SETH, planar graphs and minor-free graphs admit truly subquadratic algorithms, while geometric intersection graphs of unit balls, congruent equilateral triangles, and unit segments do not. Unit-disk graphs is one of the major open cases where the complexity of diameter computation remains unknown. More generally, it is conjectured that a truly subquadratic time algorithm exists for pseudo-disk graphs where each pair of objects has at most two intersections on the boundary. In this paper, we show a truly-subquadratic algorithm of running time O^~(n^{2-1/18}), for finding the diameter in a unit-disk graph, whose output differs from the optimal solution by at most 2. This is the first algorithm that provides an additive guarantee in distortion, independent of the size or the diameter of the graph. Our algorithm requires two important technical elements. First, we show that for the intersection graph of pseudo-disks, the graph VC-dimension - either of k-hop balls or the distance encoding vectors - is 4. This contrasts to the VC dimension of the pseudo-disks themselves as geometric ranges (which is known to be 3). Second, we introduce a clique-based r-clustering for geometric intersection graphs, which is an analog of the r-division construction for planar graphs. We also showcase the new techniques by establishing new results for distance oracles for unit-disk graphs with subquadratic storage and O(1) query time. The results naturally extend to unit L₁ or L_∞-disks and fat pseudo-disks of similar size. Last, if the pseudo-disks additionally have bounded ply, we have a truly subquadratic algorithm to find the exact diameter.