Arnold Filtser

Realistic metric spaces (such as road/transportation networks) tend to be much more algorithmically tractable than general metrics. In an attempt to formalize this intuition, Abraham et al. (SODA 2010, JACM 2016) introduced the notion of highway dimension. A weighted graph $G$ has highway dimension $h$ if for every ball $B$ of radius $\approx4r$, there is a hitting set of size $h$ hitting all the shortest paths of length $>r$ in $B$. Unfortunately, this definition fails to incorporate some very natural metric spaces such as the grid graph, and the Euclidean plane. We relax the definition of highway dimension by demanding to hit only approximate shortest paths. In addition to generalizing the original definition, this new definition also incorporates all doubling spaces (in particular the grid graph and the Euclidean plane). We then construct a PTAS for TSP under this new definition (improving a QPTAS w.r.t. the original more restrictive definition of Feldmann et al. (SICOMP 2018)). Finally, we develop a basic metric toolkit for spaces with small highway dimension by constructing padded decompositions, sparse covers/partitions, and tree covers. An abundance of applications follow. 57 pages. This is the TheoretiCS journal version

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

Given a weighted digraph $G=(V,E,w)$, a stochastic embedding into DAGs is a distribution $\mathcal{D}$ over pairs of DAGs $(D_1,D_2)$ such that for every $u,v$: (1) the reachability is preserved: $u\rightsquigarrow_G v$ (i.e., $v$ is reachable from $u$ in $G$) implies that $u\rightsquigarrow_{D_1} v$ or $u\rightsquigarrow_{D_2} v$ (but not both), and (2) distances are dominated: $d_G(u,v)\le\min\{d_{D_1}(u,v),d_{D_2}(u,v)\}$. The stochastic embedding $\mathcal{D}$ has expected distortion $t$ if for every $u,v\in V$, \[ \mathbb{E}_{(D_{1},D_{2})\sim\mathcal{D}}\left[d_{D_{1}}(u,v)\cdot\boldsymbol{1}[u\rightsquigarrow_{D_{1}}v]+d_{D_{2}}(u,v)\cdot\boldsymbol{1}[u\rightsquigarrow_{D_{2}}v]\right]\le t\cdot d_{G}(u,v)~. \] Finally, the sparsity of $\mathcal{D}$ is the maximum number of edges in any of the DAGs in its support. Given an $n$ vertex digraph with $m$ edges, we construct a stochastic embedding into DAGs with expected distortion $\tilde{O}(\log n)$ and $\tilde{O}(m)$ sparsity, improving a previous result by Assadi, Hoppenworth, and Wein [STOC 25], which achieved expected distortion $\tilde{O}(\log^3 n)$. Further, we can sample DAGs from this distribution in $\tilde{O}(m)$ time.

We study efficient algorithms for the Euclidean $k$-Center problem, focusing on the regime of large $k$. We take the approach of data reduction by considering $α$-coreset, which is a small subset $S$ of the dataset $P$ such that any $β$-approximation on $S$ is an $(α+ β)$-approximation on $P$. We give efficient algorithms to construct coresets whose size is $k \cdot o(n)$, which immediately speeds up existing approximation algorithms. Notably, we obtain a near-linear time $O(1)$-approximation when $k = n^c$ for any $0 < c < 1$. We validate the performance of our coresets on real-world datasets with large $k$, and we observe that the coreset speeds up the well-known Gonzalez algorithm by up to $4$ times, while still achieving similar clustering cost. Technically, one of our coreset results is based on a new efficient construction of consistent hashing with competitive parameters. This general tool may be of independent interest for algorithm design in high dimensional Euclidean spaces.

It was conjectured by Gupta et al. that every planar graph can be embedded into \(\ell_{1}\) with constant distortion. However, given an \(n\) -vertex weighted planar graph, the best upper bound on the distortion is only \(O(\sqrt{\log n})\) , by Rao. In this article, we study the case where there is a set \(K\) of terminals, and the goal is to embed only the terminals into \(\ell_{1}\) with low distortion. In a seminal article, Okamura and Seymour showed that if all the terminals lie on a single face, they can be embedded isometrically into \(\ell_{1}\) . The more general case, where the set of terminals can be covered by \(\gamma\) faces, was studied by Lee and Sidiropoulos and Chekuri et al. The state of the art is an upper bound of \(O(\log\gamma)\) by Krauthgamer, Lee and Rika. Our contribution is a further improvement on the upper bound to \(O(\sqrt{\log\gamma})\) . Since every planar graph has at most \(O(n)\) faces, any further improvement on this result will be a major breakthrough, directly improving upon Rao's long standing upper bound. Moreover, it is well known that the flow-cut gap equals to the distortion of the best embedding into \(\ell_{1}\) . Therefore, our result provides a polynomial time \(O(\sqrt{\log\gamma})\) -approximation to the sparsest cut problem on planar graphs, for the case where all the demand pairs can be covered by \(\gamma\) faces.

A partition \(\mathcal{P}\) of a weighted graph \(G\) is \((\sigma,\tau,\Delta)\) -sparse if every cluster has diameter at most \(\Delta\) , and every ball of radius \(\Delta/\sigma\) intersects at most \(\tau\) clusters. Similarly, \(\mathcal{P}\) is \((\sigma,\tau,\Delta)\) -scattering if instead for balls, we require that every shortest path of length at most \(\Delta/\sigma\) intersects at most \(\tau\) clusters. Given a graph \(G\) that admits a \((\sigma,\tau,\Delta)\) -sparse partition for all \(\Delta > 0\) , Jia et al. constructed a solution for the Universal Steiner Tree problem (and also Universal TSP) with stretch \(O(\tau\sigma^{2} \log _{\tau}n)\) . Given a graph \(G\) that admits a \((\sigma,\tau,\Delta)\) -scattering partition for all \(\Delta > 0\) , we construct a solution for the Steiner Point Removal problem with stretch \(O(\tau^{3}\sigma^{3})\) . We then construct sparse and scattering partitions for various different graph families, receiving many new results for the Universal Steiner Tree and Steiner Point Removal problems.

A partition $\mathcal{P}$ of a weighted graph $G$ is $(\sigma,\tau,\Delta)$-sparse if every cluster has diameter at most $\Delta$, and every ball of radius $\Delta/\sigma$ intersects at most $\tau$ clusters. Similarly, $\mathcal{P}$ is $(\sigma,\tau,\Delta)$-scattering if instead for balls we require that every shortest path of length at most $\Delta/\sigma$ intersects at most $\tau$ clusters. Given a graph $G$ that admits a $(\sigma,\tau,\Delta)$-sparse partition for all $\Delta>0$, Jia et al. [STOC05] constructed a solution for the Universal Steiner Tree problem (and also Universal TSP) with stretch $O(\tau\sigma^2\log_\tau n)$. Given a graph $G$ that admits a $(\sigma,\tau,\Delta)$-scattering partition for all $\Delta>0$, we construct a solution for the Steiner Point Removal problem with stretch $O(\tau^3\sigma^3)$. We then construct sparse and scattering partitions for various different graph families, receiving many new results for the Universal Steiner Tree and Steiner Point Removal problems.