Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case [Duraj et al., 2024; Hsien-Chih Chang et al., 2024; Chan et al., 2025] where truly subquadratic-time algorithms were given for simple objects such as unit-disks and (axis-aligned) squares. However, in three or higher dimensions, there is no known truly subquadratic-time algorithm for any intersection graph of non-trivial objects, even basic ones such as unit balls or (axis-aligned) unit cubes. This was partially explained by the pioneering work of Bringmann et al. [Karl Bringmann et al., 2022] which gave several truly subquadratic lower bounds, notably for unit balls or unit cubes in 3D when the graph diameter Δ is at least Ω(log n), hinting at a pessimistic outlook for the complexity of the diameter problem in higher dimensions. In this paper, we substantially extend the landscape of diameter computation for objects in three and higher dimensions, giving a few positive results. Our highlighted findings include: 1) A truly subquadratic-time algorithm for deciding if the diameter of unit cubes in 3D is at most 3 (Diameter-3 hereafter), the first algorithm of its kind for objects in 3D or higher dimensions. Our algorithm is based on a novel connection to pseudolines, which is of independent interest. 2) A truly subquadratic time lower bound for Diameter-3 of unit balls in 3D under the Orthogonal Vector (OV) hypothesis, giving the first separation between unit balls and unit cubes in the small diameter regime. Previously, computing the diameter for both objects was known to be quadratic hard when the diameter is Ω(log n) [Karl Bringmann et al., 2022]. 3) A near-linear-time algorithm for Diameter-2 of unit cubes in 3D, generalizing the previous result for unit squares in 2D [Karl Bringmann et al., 2022]. 4) A truly subquadratic-time algorithm and lower bound for Diameter-2 and Diameter-3 of rectangular boxes (of arbitrary dimension and sizes), respectively.

It is known that any n-point set in the d-dimensional Euclidean space ℝ^d, for d = O(1), admits: 1) A (1+ε)-spanner with maximum degree Õ(ε^{-d+1}) and with lightness Õ(ε^{-d}), for any ε > 0. 2) A (1+ε)-tree cover with Õ(n ⋅ ε^{-d+1}) trees and maximum degree of O(1) in each tree. Moreover, all the parameters in these constructions are optimal: For any 2 ≤ d = O(1), there exists an n-point set in ℝ^d, for which any (1+ε)-spanner has Ω̃(n⋅ε^{-d+1}) edges and lightness Ω̃(ε^{-d}). The upper bounds for Euclidean spanners rely heavily on the spatial property of cone partitioning in ℝ^d, which does not seem to extend to the wider family of doubling metrics, i.e., metric spaces of constant doubling dimension. In doubling metrics, a simple spanner construction from two decades ago, the net-tree spanner, has Õ(n⋅ε^{-d}) edges, and it could be transformed into a spanner of maximum degree Õ(ε^{-d}) and lightness Õ(n⋅ε^{-(d+1)}) by pruning redundant edges. Moreover, a careful refinement of the net-tree spanner yields a (1+ε)-tree cover with Õ(ε^{-d}) trees. Despite a large body of work, the problem of obtaining tight bounds for spanners and tree covers in the wider family of doubling metrics has remained elusive. We resolve this problem by presenting: 1) A surprisingly simple and tight lower bound, which shows that the net-tree spanner and its pruned version are optimal with respect to all the involved parameters. 2) A new construction of (1+ε)-tree covers with Õ(n⋅ε^{-d}) trees, with maximum degree O(1) in each tree. This construction is optimal with respect to the number of trees and maximum degree.

Sparse shortcuttings of trees - equivalently, sparse 1-spanners for tree metrics with bounded hop-diameter - have been studied extensively (under different names and settings), since the pioneering works of [Andrew Chi-Chih Yao, 1982; Chazelle, 1987; Noga Alon and Baruch Schieber, 1987; Hans L. Bodlaender et al., 1994], initially motivated by applications to range queries, online tree product, and MST verification, to name a few. These constructions were also lifted from trees to other graph families using known low-distortion embedding results. The works of [Andrew Chi-Chih Yao, 1982; Chazelle, 1987; Noga Alon and Baruch Schieber, 1987; Hans L. Bodlaender et al., 1994] establish a tight tradeoff between hop-diameter and sparsity (or average degree) for tree shortcuttings and imply constant-hop shortcuttings for n-node trees with sparsity O(log^* n). Despite their small sparsity, all known constant-hop shortcuttings contain dense subgraphs (of sparsity Ω(log n)), which is a significant drawback for many applications. We initiate a systematic study of constant-hop tree shortcuttings that are "tree-like". We focus on two well-studied graph parameters that measure how far a graph is from a tree: arboricity and treewidth. Our contribution is twofold. - New upper and lower bounds for tree-like shortcuttings of trees, including an optimal tradeoff between hop-diameter and treewidth for all hop-diameter up to O(log log n). We also provide a lower bound for larger values of k, which together yield hop-diameter× treewidth = Ω((log log n)²) for all values of hop-diameter, resolving an open question of [Arnold Filtser and Hung Le, 2022; H. Le, 2023]. - Applications of these bounds, focusing on low-dimensional Euclidean and doubling metrics. A seminal work of Arya et al. [S. Arya et al., 1995] presented a (1+ε)-spanner with constant hop-diameter and sparsity O(log^* n), but with large arboricity. We show that constant hop-diameter is sufficient to achieve arboricity O(log^*{n}). Furthermore, we present a (1+ε)-stretch routing scheme in the fixed-port model with 3 hops and a local memory of O(log²n / log log n) bits, resolving an open question of [Omri Kahalon et al., 2022].

For a weighted graph $G = (V, E, w)$ and a designated source vertex $s \in V$, a spanning tree that simultaneously approximates a shortest-path tree w.r.t. source $s$ and a minimum spanning tree is called a shallow-light tree (SLT). Specifically, an $(α, β)$-SLT of $G$ w.r.t. $s \in V$ is a spanning tree of $G$ with root-stretch $α$ (preserving all distances between $s$ and the other vertices up to a factor of $α$) and lightness $β$ (its weight is at most $β$ times the weight of a minimum spanning tree of $G$). Despite the large body of work on SLTs, the basic question of whether a better approximation algorithm exists was left untouched to date, and this holds in any graph family. This paper makes a first nontrivial step towards this question by presenting two bicriteria approximation algorithms. For any $ε>0$, a set $P$ of $n$ points in constant-dimensional Euclidean space and a source $s\in P$, our first (respectively, second) algorithm returns, in $O(n \log n \cdot {\rm polylog}(1/ε))$ time, a non-Steiner (resp., Steiner) tree with root-stretch $1+O(ε\log ε^{-1})$ and weight at most $O(\mathrm{opt}_ε\cdot \log^2 ε^{-1})$ (resp., $O(\mathrm{opt}_ε\cdot \log ε^{-1})$), where $\mathrm{opt}_ε$ denotes the minimum weight of a non-Steiner (resp., Steiner) tree with root-stretch $1+ε$.

Computing the diameter of the intersection graphs of objects is a basic problem in computational geometry. Previous works showed that the complexity of computing the diameter mainly depends on the object types: for unit disks and squares in 2D, the problem is solvable in truly subquadratic time [Chan et al., 2025], while for other objects, including unit segments and equilateral triangles in 2D or unit balls and axis-parallel unit cubes in 3D, there is no truly subquadratic time algorithm under the Orthogonal Vector (OV) hypothesis [Bringmann et al., 2022]. We undertake a comprehensive study of computing the diameter of geometric intersection graphs for various types of objects. We discover many new irregularities, showing that the landscape is extremely nuanced: the source of hardness is a combination of the object type, the true diameter value, and how the objects intersect with each other. Our highlighted results for the 2D case include: 1) The diameter of non-degenerate, axis-aligned line segments can be computed in truly subquadratic time. Previous hardness result [Bringmann et al., 2022] for line segments applies only to degenerate instances. On the other hand, for the degenerate case, we show that a truly subquadratic time algorithm exists when the true diameter is constant. 2) An almost-linear-time algorithm for unit-square graphs of constant diameter. Previous algorithms [Duraj et al., 2024; Chan et al., 2025] rely on succinct representation assuming bounded VC-dimension; for such a strategy Ω(n^{7/4}) time is an inherent barrier. 3) An Õ(n^{4/3})-time algorithm to decide if the diameter of a unit-disk graph is at most 2. This improves upon the recent algorithm with running time Õ(n^{2-1/9}) [Chan et al., 2025]. 4) Deciding if the diameter of intersection graphs of fat triangles or line segments is at most 2 is truly subquadratic-hard under fine-grained complexity assumptions. Previous lower bounds [Bringmann et al., 2022] only hold when deciding if diameter is at most 3. Our findings are presented in a pair of papers. This paper focuses solely on the 2D case, while the companion paper is devoted to higher-dimensional cases.

A \(t\)-stretch tree cover of a metric space \(M = (X, \delta)\), for a parameter \(t \ge 1\), is a collection of trees such that every pair of points has a \(t\)-stretch path in one of the trees. Tree covers provide an important sketching tool that has found various applications over the years. The celebrated Dumbbell Theorem by Arya et al. [STOC’95] states that any set of points in the Euclidean plane admits a \((1+\epsilon)\)-stretch tree cover with \(O_\epsilon(1)\) trees. This result extends to any (constant) dimension and was also generalized for arbitrary doubling metrics by Bartal et al. [ICALP’19].

In their seminal paper, Althöfer et al. (DCG 1993) introduced the greedy spanner and showed that, for any weighted planar graph \(G\), the weight of the greedy \((1+\epsilon)\)-spanner is at most \((1+\tfrac{2}{\epsilon})\cdot \textrm w(\mathsf{MST}(G))\), where \(\textrm w(\mathsf{MST}(G))\) is the weight of a minimum spanning tree \(\mathrm{MST}(G)\) of \(G\). This bound is optimal in an \(\textcolor{#ff6666}{existential\ sense}\): there exist planar graphs \(G\) for which any \((1+\epsilon)\)-spanner has a weight of at least \((1+\tfrac{2}{\epsilon})\cdot w(\mathrm{MST}(G))\).

In the (continuous) Euclidean k -center problem, given n points in ℝ d and an integer k , the goal is to find k center points in ℝ d that minimize the maximum Euclidean distance from any input point to its closest center. In this paper, we establish conditional lower bounds for this problem in constant dimensions in two settings. Parameterized by k : Assuming the Exponential Time Hypothesis (ETH), we show that there is no f ( k ) n o ( k 1−1/ d ) -time algorithm for the Euclidean k -center problem. This result shows that the algorithm of Agarwal and Procopiuc [SODA 1998; Algorithmica 2002] is essentially optimal. Furthermore, our lower bound rules out any (1+ε)-approximation algorithm running in time ( k /ε) o ( k 1−1/ d ) n O (1) , thereby establishing near-optimality of the corresponding approximation scheme by the same authors. Small k : Assuming the 3-SUM hypothesis, we prove that for any ε>0 there is no O ( n 2−ε )-time algorithm for the Euclidean 2-center problem in ℝ 3 . This settles an open question posed by Agarwal, Ben Avraham, and Sharir [SoCG 2010; Computational Geometry 2013]. In addition, under the same hypothesis, we prove that for any ε > 0, the Euclidean 6-center problem in ℝ 2 also admits no O ( n 2−ε )-time algorithm. The technical core of all our proofs is a novel geometric embedding of a system of linear equations. We construct a point set where each variable corresponds to a specific collection of points, and the geometric structure ensures that a small-radius clustering is possible if and only if the system has a valid solution.

The planar separator theorem by Lipton and Tarjan [FOCS '77, SIAM Journal on Applied Mathematics '79] states that any planar graph with $n$ vertices has a balanced separator of size $O(\sqrt{n})$ that can be found in linear time. This landmark result kicked off decades of research on designing linear or nearly linear-time algorithms on planar graphs. In an attempt to generalize Lipton-Tarjan's theorem to nonplanar graphs, Alon, Seymour, and Thomas [STOC '90, Journal of the AMS '90] showed that any minor-free graph admits a balanced separator of size $O(\sqrt{n})$ that can be found in $O(n^{3/2})$ time. The superlinear running time in their separator theorem is a key bottleneck for generalizing algorithmic results from planar to minor-free graphs. Despite extensive research for more than two decades, finding a balanced separator of size $O(\sqrt{n})$ in (linear) $O(n)$ time for minor-free graphs remains a major open problem. Known algorithms either give a separator of size much larger than $O(\sqrt{n})$ or have superlinear running time, or both. In this paper, we answer the open problem affirmatively. Our algorithm is very simple: it runs a vertex-weighted variant of breadth-first search (BFS) a constant number of times on the input graph. Our key technical contribution is a weighting scheme on the vertices to guide the search for a balanced separator, offering a new connection between the size of a balanced separator and the existence of a clique-minor model. We believe that our weighting scheme may be of independent interest.

We give the first truly subquadratic time algorithm, with ${O^{\ast}}\left( {{n^{2 - 1/18}}} \right)$ running time, for computing the diameter of an n-vertex unit-disk graph, resolving a central open problem in the literature. Our result is obtained as an instance of a general framework, applicable to different graph families and distance problems. Surprisingly, our framework completely bypasses sublinear separators (or r-divisions) which were used in all previous algorithms. Instead, we use low-diameter decompositions in their most elementary form. We also exploit bounded VC-dimension of set systems associated with the input graph, as well as new ideas on geometric data structures. Among the numerous applications of the general framework, we obtain:1)An $\tilde O\left( {m{n^{1 - 1/(2d)}}} \right)$ time algorithm for computing the diameter of m-edge sparse unweighted graphs with constant VC-dimension d. The previously known algorithms by Ducoffe, Habib, and Viennot [SODA 2019] and Duraj, Konieczny, and Potępa [ESA 2024] are truly subquadratic only when the diameter is a small polynomial. Our result thus generalizes truly subquadratic time algorithms known for planar and minor-free graphs (in fact, it slightly improves the previous time bound for minor-free graphs).2)An $\tilde O\left( {{n^{2 - 1/12}}} \right)$ time algorithm for computing the diameter of intersection graphs of axis-aligned squares with arbitrary size. The best-known algorithm by Duraj, Konieczny, and Potępa [ESA 2024] only works for unit squares and is only truly subquadratic in the low-diameter regime.3)The first algorithms with truly subquadratic complexity for other distance-related problems, including all-vertex eccentricities, Wiener index, and exact distance oracles. In particular, we obtain the first exact distance oracle with truly subquadratic space and $\tilde O(1)$ query time for any sparse graph with bounded VC-dimension, again generalizing previous results for planar and minor-free graphs.

We study spanners in planar domains, including polygonal domains, polyhedral terrain, and planar metrics. Previous work showed that for any constant ε ∈ (0,1), one could construct a (2 + ε )-spanner with O (n log(n )) edges (SICOMP 2019), and there is a lower bound of Ω(n2) edges for any (2 — ε )-spanner (SoCG 2015). The main open question is whether a linear number of edges suffices and the stretch can be reduced to 2. We resolve this problem by showing that for stretch 2, one needs Ω(n log n ) edges, and for stretch 2 + ε for any fixed ε ∈ (0,1), O (n ) edges are sufficient. Our lower bound is the first super-linear lower bound for stretch 2.

Cohen-Addad, Le, Pilipczuk, and Pilipczuk [CLPP23] recently constructed a stochastic embedding with expected 1 + ε distortion of n-vertex planar graphs (with polynomial aspect ratio) into graphs of treewidth O (ε-1 log13n ). Their embedding is the first to achieve polylogarithmic treewidth. However, there remains a large gap between the treewidth of their embedding and the treewidth lower bound of Ω(log n ) shown by Carroll and Goel [CG04]. In this work, we substantially narrow the gap by constructing a stochastic embedding with treewidth O (ε-1 log3 n ).

Abstract. Euclidean spanners are important geometric structures, having found numerous applications over the years. Cornerstone results in this area from the late 1980s and early 1990s state that for any [Formula: see text]-dimensional [Formula: see text]-point Euclidean space, there exists a [Formula: see text]-spanner with [Formula: see text] edges and lightness (normalized weight) [Formula: see text]. Surprisingly, the fundamental question of whether or not these dependencies on [Formula: see text] and [Formula: see text] for small [Formula: see text] can be improved has remained elusive, even for [Formula: see text]. This question naturally arises in any application of Euclidean spanners where precision is a necessity (thus [Formula: see text] is tiny). In the most extreme case [Formula: see text] is inverse polynomial in [Formula: see text], and then one could potentially improve the size and lightness bounds by factors that are polynomial in [Formula: see text]. The state-of-the-art bounds [Formula: see text] and [Formula: see text] on the size and lightness of spanners are realized by the greedy spanner. In 2016, in a preliminary version, Filtser and Solomon [ SIAM J. Comput., 49 (2020), pp. 429–447] proved that, in low-dimensional spaces, the greedy spanner is “near-optimal”; informally, their result states that the greedy spanner for dimension [Formula: see text] is just as sparse and light as any other spanner but for dimension larger by a constant factor. Hence the question of whether the greedy spanner is truly optimal remained open to date. The contribution of this paper is twofold: (1) We resolve these longstanding questions by nailing down the dependencies on [Formula: see text] and [Formula: see text] and showing that the greedy spanner is truly optimal. Specifically, for any [Formula: see text], (a) we show that there are [Formula: see text]-point sets in [Formula: see text] for which any [Formula: see text]-spanner must have [Formula: see text] edges, implying that the greedy (and other) spanners achieve the optimal size; (b) we show that there are [Formula: see text]-point sets in [Formula: see text] for which any [Formula: see text]-spanner must have lightness [Formula: see text], and then improve the upper bound on the lightness of the greedy spanner from [Formula: see text] to [Formula: see text]. (The lightness upper and lower bounds match up to a lower-order term.) (2) We then complement our negative result for the size of spanners with a rather counterintuitive positive result: Steiner points lead to a quadratic improvement in the size of spanners! Our bound for the size of Steiner spanners in [Formula: see text] is tight as well (up to a lower-order term).

Abstract. We present a unified framework for constructing light spanners in a variety of graph classes. Informally, the framework boils down to a transformation from sparse spanners to light spanners; since the state-of-the-art for sparse spanners is much more advanced than that for light spanners, such a transformation is powerful. Our framework is developed in two papers. The current paper is the first of the two—it lays the basis of the unified framework and then applies it to design fast constructions with optimal lightness for several graph classes. Our new constructions are significantly faster than the state-of-the-art for every graph class studied in this paper; the running times of our constructions are near-linear and usually optimal. Among various applications and implications of our framework, we highlight here the following (for simplicity assume [Formula: see text] is fixed): (1) In low-dimensional Euclidean spaces, we present a construction of [Formula: see text]-spanners for [Formula: see text]-point sets with lightness and degree both bounded by constants, running in [Formula: see text] time in the algebraic computation tree (ACT) (or real-RAM) model, which is the basic model used in computational geometry. Our construction is optimal with respect to all the involved quality measures—running time, lightness, and degree—and it resolves a major problem in the area of geometric spanners, which was open for three decades. (2) In general graphs, we present a near-linear time algorithm for constructing light spanners of graphs with [Formula: see text] vertices and [Formula: see text] edges. Specifically, for any [Formula: see text], we construct a [Formula: see text]-spanner with lightness [Formula: see text] in [Formula: see text] time, where [Formula: see text] is the inverse Ackermann function; the lightness bound matches Erdős’ girth conjecture up to the [Formula: see text]-dependency. (Our companion paper builds on the basis laid in this paper, aiming to achieve optimality in a more refined sense, which takes into account a wider range of involved parameters, most notably [Formula: see text], but also others such as the Euclidean dimension or the minor size (in minor-free graphs).)

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.

A $(1+\varepsilon)\textit{-stretch tree cover}$ of a metric space is a collection of trees, where every pair of points has a $(1+\varepsilon)$-stretch path in one of the trees. The celebrated $\textit{Dumbbell Theorem}$ [Arya et~al. STOC'95] states that any set of $n$ points in $d$-dimensional Euclidean space admits a $(1+\varepsilon)$-stretch tree cover with $O_d(\varepsilon^{-d} \cdot \log(1/\varepsilon))$ trees, where the $O_d$ notation suppresses terms that depend solely on the dimension~$d$. The running time of their construction is $O_d(n \log n \cdot \frac{\log(1/\varepsilon)}{\varepsilon^{d}} + n \cdot \varepsilon^{-2d})$. Since the same point may occur in multiple levels of the tree, the $\textit{maximum degree}$ of a point in the tree cover may be as large as $Ω(\log Φ)$, where $Φ$ is the aspect ratio of the input point set. In this work we present a $(1+\varepsilon)$-stretch tree cover with $O_d(\varepsilon^{-d+1} \cdot \log(1/\varepsilon))$ trees, which is optimal (up to the $\log(1/\varepsilon)$ factor). Moreover, the maximum degree of points in any tree is an $\textit{absolute constant}$ for any $d$. As a direct corollary, we obtain an optimal {routing scheme} in low-dimensional Euclidean spaces. We also present a $(1+\varepsilon)$-stretch $\textit{Steiner}$ tree cover (that may use Steiner points) with $O_d(\varepsilon^{(-d+1)/{2}} \cdot \log(1/\varepsilon))$ trees, which too is optimal. The running time of our two constructions is linear in the number of edges in the respective tree covers, ignoring an additive $O_d(n \log n)$ term; this improves over the running time underlying the Dumbbell Theorem.

Euclidean spanners are important geometric objects that have been extensively studied since the 1980s. The two most basic “compactness” measures of a Euclidean spanner$E$12We shall identify a graph$H = (X,E)$with its edge set$E$. All edge weights are given by the Euclidean distances. are the size (number of edges)$\vert E\vert$and the weight (sum of edge weights)$\Vert E\Vert$. The state-of-the-art constructions of Euclidean$(1+\epsilon)$-spanners in$\mathbb{R}^{d}$have$o_{d}(_{n\cdot\epsilon^{-d+1}})$edges (or sparsity$O_{d}(\epsilon^{-d+1}))$and weight$O_{d}(\epsilon^{-d} \log \epsilon^{-1}) \cdot\Vert E_{\text{mst}}\Vert$(or lightness$O_{d}(\epsilon^{-d}\log\epsilon^{-1}))$; here$O_{d}$suppresses a factor of$d^{O(d)}$and$\Vert E_{\text{mst}}\Vert$denotes the weight of a minimum spanning tree of the input point set. Importantly, these two upper bounds are (near-)optimal (up to the$d^{O(d)}$factor and disregarding the factor of$\log(\epsilon^{-1})$in the lightness bound) for some extremal instances [Le and Solomon, 2019], and therefore they are (near-)optimal in an existential sense. Moreover, both these upper bounds are attained by the same construction-the classic greedy spanner, whose sparsity and lightness are not only existentially optimal, but they also significantly outperform those of any other Euclidean spanner construction studied in an experimental study by [Farshi-Gudmundsson, 2009] for various practical point sets in the plane. This raises the natural question of whether the greedy spanner is (near-) optimal for any point set instance? Motivated by this question, we initiate the study of instance optimal Euclidean spanners. Our results are two-fold. •Rather surprisingly (given the aforementioned experimental study), we demonstrate that the greedy spanner is far from being instance optimal, even when allowing its stretch to grow. More concretely, we design two hard instances of point sets in the plane, where the greedy$(1+x\epsilon)$-spanner (for basically any parameter$x \geq 1$) has$\Omega_{x}(\epsilon^{-1/2})\cdot\vert E_{\text{spa}} \vert$edges and weight$\Omega_{x}(\epsilon^{-1})\cdot\Vert E_{\text{light}}\Vert$, where$E_{\text{spa}}$and$E_{\text{light}}$denote the per-instance sparsest and lightest$(1 +\epsilon)$-spanners, respectively, and the$\Omega_{x}$notation suppresses a polynomial dependence on$1/x$. •As our main contribution, we design a new construction of Euclidean spanners, which is inherently different from known constructions, achieving the following bounds: a stretch of$1+\epsilon\cdot 2^{O(\log^{*}(d/\epsilon)}$with$O(1)\cdot\vert E_{\text{spa}}\vert$edges and weight$O(1)$. $\Vert E_{ \text{light}}\Vert$. In other words, we show that a slight increase to the stretch suffices for obtaining instance optimality up to an absolute constant for both sparsity and lightness. Remarkably, there is only a log-star dependence on the dimension in the stretch, and there is no dependence on it whatsoever in the number of edges and weight. In general, for any integer$k\geq 1$, we can construct a Euclidean spanner in$\mathbb{R}^{d}$of stretch$1+\epsilon\cdot 2^{O(k)}$with$O(\log^{(k)}(\epsilon^{-1})+\log^{(k-1)}(d))\cdot\vert E_{\text{spa}}\vert$edges and weight$O(\log^{(k)}(\epsilon^{-1})+\log^{(k-1)}(d))\cdot\Vert E_{\text{light}}\Vert$, where$\log^{(k)}$denotes the k-iterated logarithm.

The notion of shortcut partition, introduced recently by Chang, Conroy, Le, Milenković, Solomon, and Than [CCL+23], is a new type of graph partition into low-diameter clusters. Roughly speaking, the shortcut partition guarantees that for every two vertices u and v in the graph, there exists a path between u and v that intersects only a few clusters. They proved that any planar graph admits a shortcut partition and gave several applications, including a construction of tree cover for arbitrary planar graphs with stretch 1 + ɛ and O(1) many trees for any fixed ɛ ∈ (0,1). However, the construction heavily exploits planarity in multiple steps, and is thus inherently limited to planar graphs.

A recent line of work on VC set systems in minor-free (undirected) graphs, starting from Li and Parter [LP19], who constructed a new VC set system for planar graphs, has given surprising algorithmic results [LP19, Le23, DHV20, FHMWN20]. In this work, we initialize a more systematic study of VC set systems for minor-free graphs and their applications in both undirected graphs and directed graphs (a.k.a digraphs). More precisely:

The greedy spanner in a low-dimensional Euclidean space is a fundamental geometric construction that has been extensively studied over three decades, as it possesses the two most basic properties of a good spanner: constant maximum degree and constant lightness. Recently, Eppstein and Khodabandeh [ 28 ] showed that the greedy spanner in \(\mathbb {R}^2\) admits a sublinear separator in a strong sense: Any subgraph of k vertices of the greedy spanner in \(\mathbb {R}^2\) has a separator of size \(O(\sqrt {k})\) . Their technique is inherently planar and is not extensible to higher dimensions. They left showing the existence of a small separator for the greedy spanner in \(\mathbb {R}^d\) for any constant d ≥ 3 as an open problem. In this article, we resolve the problem of Eppstein and Khodabandeh [ 28 ] by showing that any subgraph of k vertices of the greedy spanner in \(\mathbb {R}^d\) has a separator of size \(O(k^{1-1/d})\) . We introduce a new technique that gives a simple criterion for any geometric graph to have a sublinear separator that we dub τ-lanky : A geometric graph is τ-lanky if any ball of radius r cuts at most τ edges of length at least r in the graph. We show that any τ-lanky geometric graph of n vertices in \(\mathbb {R}^d\) has a separator of size \(O(\tau n^{1-1/d})\) . We then derive our main result by showing that the greedy spanner is O (1)-lanky. We indeed obtain a more general result that applies to unit ball graphs and point sets of low fractal dimensions in \(\mathbb {R}^d\) . Our technique naturally extends to doubling metrics. We use the τ-lanky criterion to show that there exists a (1+ε)-spanner for doubling metrics of dimension d with a constant maximum degree and a separator of size \(O(n^{1-\frac{1}{d}})\) ; this result resolves an open problem posed by Abam and Har-Peled [ 1 ] a decade ago. We then introduce another simple criterion for a graph in doubling metrics of dimension d to have a sublinear separator. We use the new criterion to show that the greedy spanner of an n -point metric space of doubling dimension d has a separator of size \(O((n^{1-\frac{1}{d}}) + \log \Delta)\) where Δ is the spread of the metric; the factor log (Δ) is tightly connected to the fact that, unlike its Euclidean counterpart, the greedy spanner in doubling metrics has unbounded maximum degree . Finally, we discuss algorithmic implications of our results.

While research on the geometry of planar graphs has been active in the past decades, many properties of planar metrics remain mysterious. This paper studies a fundamental aspect of the planar graph geometry: covering planar metrics by a small collection of simpler metrics. Specifically, a tree cover of a metric space $(X, \delta)$ is a collection of trees, so that every pair of points u and v in X has a low-distortion path in at least one of the trees.The celebrated “Dumbbell Theorem” [ADM+95] states that any low-dimensional Euclidean space admits a tree cover with $O(1)$ trees and distortion $1+\varepsilon$, for any fixed $\varepsilon \in(0,1)$. This result has found numerous algorithmic applications, and has been generalized to the wider family of doubling metrics [BFN19]. Does the same result hold for planar metrics? A positive answer would add another evidence to the well-observed connection between Euclidean/doubling metrics and planar metrics.In this work, we answer this fundamental question affirmatively. Specifically, we show that for any given fixed $\varepsilon \in(0,1)$, any planar metric can be covered by $O(1)$ trees with distortion $1+\varepsilon$. Our result for planar metrics follows from a rather general framework: First we reduce the problem to constructing tree covers with additive distortion. Then we introduce the notion of shortcut partition, and draw connection between shortcut partition and additive tree cover. Finally we prove the existence of shortcut partition for any planar metric, using new insights regarding the grid-like structure of planar graphs. To demonstrate the power of our framework:•We establish additional tree cover results beyond planar metrics; in particular, we present an $O(1)$-size tree cover with distortion $1+\varepsilon$ for bounded treewidth metrics;•We obtain several algorithmic applications in planar graphs from our tree cover. The grid-like structure is a technical contribution that we believe is of independent interest. We showcase its applicability beyond tree cover by constructing a simpler and better embedding of planar graphs into $O(1)$-treewidth graphs with small additive distortion, resolving an open problem in this line of research.

We prove that there is a randomized polynomialtime algorithm that given an edge-weighted graph G excluding a fixed-minor Q on n vertices and an accuracy parameter $\varepsilon\gt$ 0, constructs an edge-weighted graph H and an embedding $\eta: V(G) \rightarrow V(H)$ with the following properties:•For any constant size Q, the treewidth of H is polynomial in $\varepsilon^{-1}, \log n$, and the logarithm of the stretch of the distance metric in G.•The expected multiplicative distortion is $(1+\varepsilon)$: for every pair of vertices $u, v$ of G, we have $\operatorname{dist}_{H}(\eta(u), \eta(v)) \geqslant \operatorname{dist}_{G}(u, v)$ always and $\mathbb{E}\left[\operatorname{dist}_{H}(\eta(u), \eta(v))\right] \leqslant(1+\varepsilon) \operatorname{dist}_{G}(u, v)$. Our embedding is the first to achieve polylogarithmic treewidth of the host graph and comes close to the lower bound by Carroll and Goel, who showed that any embedding of a planar graph with $\mathcal{O}(1)$ expected distortion requires the host graph to have treewidth $\Omega(\log n)$. It also provides a unified framework for obtaining randomized quasi-polynomial-time approximation schemes for a variety of problems including network design, clustering or routing problems, in minor-free metrics where the optimization goal is the sum of selected distances. Applications include the capacitated vehicle routing problem, and capacitated clustering problems.

An essential requirement of spanners in many applications is to be fault-tolerant: a $(1+\epsilon)$-spanner of a metric space is called (vertex) f-fault-tolerant $(f-F T)$ if it remains a $(1+\epsilon)$-spanner (for the non-faulty points) when up to f faulty points are removed from the spanner. Fault-tolerant (FT) spanners for Euclidean and doubling metrics have been extensively studied since the 90 s. For low-dimensional Euclidean metrics, Czumaj and Zhao in SoCG’03 [CZ03] showed that the optimal guarantees $O(f n), O(f)$ and $O\left(f^{2}\right)$ on the size, degree and lightness of f-FT spanners can be achieved via a greedy algorithm, which naïvely runs in $O\left(n^{3}\right) \cdot 2^{O(f)}$ time.$^{1}$ An earlier construction, by Levcopoulos et al. [LNS98] from STOC’98, has a faster running time of $O(n \log n)+n 2^{O(f)}$, but has a slack of $2^{\Omega(f)}$ in all the three involved parameters. The question of whether the optimal bounds of [CZ03] can be achieved via a fast construction has remained elusive, with the lightness parameter being the bottleneck: Any construction (other than [CZ03]) has lightness either $2^{\Omega(f)}$ or $\Omega(\log n)$. Moreover, in the wider family of doubling metrics, it is not even clear whether there exists an f FT spanner with lightness that depends solely on f (even exponentially): all existing constructions have lightness $\Omega(\log n)$ since they are built on the net-tree spanner, which is induced by a hierarchical net-tree of lightness $\Omega(\log n)$. In this paper we settle in the affirmative these longstanding open questions. Specifically, we design a construction of f-FT spanners that is optimal with respect to all the involved parameters (size, degree, lightness and running time): For any n-point doubling metric, any $\epsilon\gt0$, and any integer $1 \le \log f n\le + {nfn}-)$, an2, our construction provides,-spanner with within time $O(n$ size $O (fn)$, degree $O(f)$ and lightness $O(f^{2})$. To break the $\Omega (\log n)$ lightness barrier, we introduce a new geometric object — the light net-forest. Like the net-tree, the light net-forest is induced by a hierarchy of nets. However, to ensure small lightness, the light net-forest is inherently less “well-connected” than the net-tree, which, in turn, makes the task of achieving fault-tolerance significantly more challenging. Further, to achieve the optimal degree (and size) together with optimal lightness, and to do so within the optimal running time — we overcome several highly nontrivial technical challenges.

Thorup [FOCS'01, JACM'04] and Klein [SODA'01] independently showed that there exists a (1 + ε)-approximate distance oracle for planar graphs with O(n(log n)ε-1) space and O(ε-1) query time. While the dependency on n is nearly linear, the space-query product of their oracles depend quadratically on 1/ε. Many follow-up results either improved the space or the query time of the oracles while having the same, sometimes worst, dependency on 1/ε. Kawarabayashi, Sommer, and Thorup [SODA'13] were the first to improve the dependency on 1/ε from quadratic to nearly linear (at the cost of log*(n) factors). It is plausible to conjecture that the linear dependency on 1/ε is optimal: for many known distance-related problems in planar graphs, it was proved that the dependency on 1/ε is at least linear.

Seminal works on light spanners over the years provide spanners with optimal lightness in various graph classes, such as in general graphs, Euclidean spanners, and minor-free graphs. Three shortcomings of previous works on light spanners are: (i) The runtimes of these constructions are almost always sub-optimal, and usually far from optimal. (ii) These constructions are optimal in the standard and crude sense, but not in a refined sense that takes into account a wider range of involved parameters. (iii) The techniques are ad hoc per graph class, and thus can’t be applied broadly.

In STOC'95 [ADMSS95] Arya et al. showed that any set of n points in R^d admits a (1+ε)-spanner with hop-diameter at most 2 (respectively, 3) and O(n log n) edges (resp., O(n log log n) edges). They also gave a general upper bound tradeoff of hop-diameter at most k and O(n α_k(n)) edges, for any k≥2. The function α_k is the inverse of a certain Ackermann-style function at the ⌊k/2⌋th level of the primitive recursive hierarchy, where α₀(n)=⌈n/2⌉, α₁(n)=⌈√n⌉, α₂(n)=⌈log n⌉, α₃(n)=⌈log log n⌉, α₄(n)=log^* n, α₅(n)=⌊1/2 log^*n⌋, .... Roughly speaking, for k≥2 the function α_{k} is close to ⌊(k-2)/2⌋-iterated log-star function, i.e., log with ⌊(k-2)/2⌋ stars. Also, α_{2α(n)+4}(n)≤4, where α(n) is the one-parameter inverse Ackermann function, which is an extremely slowly growing function. Whether or not this tradeoff is tight has remained open, even for the cases k=2 and k=3. Two lower bounds are known: The first applies only to spanners with stretch 1 and the second is sub-optimal and applies only to sufficiently large (constant) values of k. In this paper we prove a tight lower bound for any constant k: For any fixed ε>0, any (1+ε)-spanner for the uniform line metric with hop-diameter at most k must have at least Ω(n α_k(n)) edges.

Cohen-Addad, Filtser, Klein and Le [FOCS’20] constructed a stochastic embedding of minor-free graphs of diameter D into graphs of treewidth $O_{\epsilon}(\log n)$ with expected additive distortion $+\epsilon D$. Cohen-Addad et al. then used the embedding to design the first quasi-polynomial time approximation scheme (QPTAS) for the capacitated vehicle routing problem. Filtser and Le [STOC’21] used the embedding (in a different way) to design a QPTAS for the metric Baker’s problems in minor-free graphs. In this work, we devise a new embedding technique to improve the treewidth bound of Cohen-Addad et al. exponentially to $O_{\epsilon}(\log \log n)^{2}$. As a corollary, we obtain the first efficient PTAS for the capacitated vehicle routing problem in minor-free graphs. We also significantly improve the running time of the QPTAS for the metric Baker’s problems in minor-free graphs from $n^{O_{\epsilon}(\log (n))}$ to $n^{O_{\epsilon}(\log \log (n))^{3}}$. Applying our embedding technique to planar graphs, we obtain a deterministic embedding of planar graphs of diameter D into graphs of treewidth $\left.O\left((\log \log n)^{2}\right) / \epsilon\right)$ and additive distortion $+\epsilon D$ that can be constructed in nearly linear time. Important corollaries of our result include a bicriteria PTAS for metric Baker’s problems and a PTAS for the vehicle routing problem with bounded capacity in planar graphs, both run in almost-linear time. The running time of our algorithms is significantly better than previous algorithms that require quadratic time. A key idea in our embedding is the construction of an (exact) emulator for tree metrics with treewidth $O(\log \log n)$ and hop-diameter $O(\log \log n)$. This result may be of independent interest.

Spanners for metric spaces have been extensively studied, perhaps most notably in low-dimensional Euclidean spaces - due to their numerous applications. Euclidean spanners can be viewed as means of compressing the (n2) pairwise distances of a d-dimensional Euclidean space into O(n) = O∈,d (n) spanner edges, so that the spanner distances preserve the original distances to within a factor of 1 + ε, for any ε > 0. Moreover, one can compute such spanners efficiently in the standard centralized and distributed settings. Once the spanner has been computed, it serves as a "proxy" overlay network, on which the computation can proceed, which gives rise to huge savings in space and other important quality measures. The original metric enables us to "navigate" optimally - a single

Let G = (V, E, w) be a weighted undirected graph on |V| = n vertices and |E| = m edges, let k ≥ 1 be any integer, and let ∊ < 1 be any parameter. We present the following results on fast constructions of spanners with near-optimal sparsity and lightness,1 which culminate a long line of work in this area. (By near-optimal we mean optimal under Erdos' girth conjecture and disregarding the ∊-dependencies.) There are (deterministic) algorithms for constructing (2k–1)(1 + ∊)-spanners for G with a near-optimal sparsity of O(n1/k · log(1/∊)/∊)). The first algorithm can be implemented in the pointer-machine model within time O(mα(m, n) · log(1/∊)/∊)+ SORT(m)), where α(·,·) is the two-parameter inverse-Ackermann function and SORT(m) is the time needed to sort m integers. The second algorithm can be implemented in the Word RAM model within time O(m log(1/∊)/∊)). There is a (deterministic) algorithm for constructing a (2k–1)(1 + ∊)-spanner for G that achieves a near-optimal bound of O(n1/k ·poly(1/∊)) on both sparsity and lightness. This algorithm can be implemented in the pointer-machine model within time O(mα(m,n) · poly(1/∊) + SORT(m)) and in the Word RAM model within time O(mα(m,n) · poly(1/∊)). The previous fastest constructions of (2k–1)(1 + ∊)-spanners with near-optimal sparsity incur a runtime of is O(min{m(n1+1/k) + n log n, k · n2+1/k}), even regardless of the lightness. Importantly, the greedy spanner for stretch 2k–1 has sparsity O(n1/k) — with no ∊-dependence whatsoever, but its runtime is O(m(n1+1/k + n log n)). Moreover, the state-of-the-art lightness bound of any (2k–1)-spanner (including the greedy spanner) is poor, even regardless of the sparsity and runtime.

The greedy spanner in a low dimensional Euclidean space is a fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of a good spanner: constant maximum degree and constant lightness. Recently, Eppstein and Khodabandeh [EK21] showed that the greedy spanner in ℝ2 admits a sublinear separator in a strong sense: any subgraph of k vertices of the greedy spanner in ℝ2 has a separator of size . Their technique is inherently planar and is not extensible to higher dimensions. They left showing the existence of a small separator for the greedy spanner in ℝd for any constant d ≥ 3 as an open problem. In this paper, we resolve the problem of Eppstein and Khodabandeh [EK21] by showing that any subgraph of k vertices of the greedy spanner in ℝd has a separator of size O(k1–1/d). We introduce a new technique that gives a simple criterion for any geometric graph to have a sublinear separator that we dub τ-lanky: a geometric graph is τ-lanky if any ball of radius r cuts at most τ edges of length at least r in the graph. We show that any τ-lanky geometric graph of n vertices in ℝd has a separator of size O(τn1–1/d). We then derive our main result by showing that the greedy spanner is O(1)-lanky. We indeed obtain a more general result that applies to unit ball graphs and point sets of low fractal dimensions in ℝd. Our technique naturally extends to doubling metrics. We use the τ-lanky criterion to show that there exists a (1 + ∊)-spanner for doubling metrics of dimension d with a constant maximum degree and a separator of size ; this result resolves an open problem posed by Abam and Har-Peled [AHP10] a decade ago. We then introduce another simple criterion for a graph in doubling metrics of dimension d to have a sublinear separator. We use the new criterion to show that the greedy spanner of an n-point metric space of doubling dimension d has a separator of size where Δ is the spread of the metric; the factor log(Δ) is tightly connected to the fact that, unlike its Euclidean counterpart, the greedy spanner in doubling metrics has unbounded maximum degree. Finally, we discuss algorithmic implications of our results.

Chan, Har-Peled, and Jones [2020] recently developed locality-sensitive ordering (LSO), a new tool that allows one to reduce problems in the Euclidean space ℝd to the 1-dimensional line. They used LSO’s to solve a host of problems. Later, Buchin, Har-Peled, and Oláh [2019,2020] used the LSO of Chan et al. to construct very sparse reliable spanners for the Euclidean space. A highly desirable feature of a reliable spanner is its ability to withstand a massive failure: the network remains functioning even if 90% of the nodes fail. In a follow-up work, Har-Peled, Mendel, and Oláh [2021] constructed reliable spanners for general and topologically structured metrics. Their construction used a different approach, and is based on sparse covers.

A ($1+\epsilon$) -approximate distance oracle of an edge-weighted graph is a data structure that returns an approximate shortest path distance between any two query vertices up to a ($1+\epsilon$) factor. Thorup (FOCS 2001, JACM 2004) and Klein (SODA 2002) independently constructed a ($1+\epsilon$) -approximate distance oracle with$O(n\log n)$space, measured in number of words, and$O(1)$query time when$G$is an undirected planar graph with$n$vertices and$\epsilon$is a fixed constant. Many follow-up works gave ($1+\epsilon$) -approximate distance oracles with various trade-offs between space and query time. However, improving$O(n\log n)$space bound without sacrificing query time remains an open problem for almost two decades. In this work, we resolve this problem affirmatively by constructing a ($1+\epsilon$) approximate distance oracle with optimal$O(n)$space and$O(1)$query time for undirected planar graphs and fixed$\epsilon$. We also make substantial progress for planar digraphs with non-negative edge weights. For fixed$\epsilon > 0$, we give a ($1+\epsilon$) -approximate distance oracle with space$o(n\log(Nn))$and$O(\log\log(Nn)$query time; here$N$is the ratio between the largest and smallest positive edge weight. This improves Thorup's (FOCS 2001, JACM 2004)$O(n\log(Nn)\log n)$space bound by more than a logarithmic factor while matching the query time of his structure. This is the first improvement for planar digraphs in two decades, both in the weighted and unweighted setting.

In low distortion metric embeddings, the goal is to embed a host “hard” metric space into a “simpler” target space while approximately preserving pairwise distances. A highly desirable target space is that of a tree metric. Unfortunately, such embedding will result in a huge distortion. A celebrated bypass to this problem is stochastic embedding with logarithmic expected distortion. Another bypass is Ramsey-type embedding, where the distortion guarantee applies only to a subset of the points. However, both these solutions fail to provide an embedding into a single tree with a worst-case distortion guarantee on all pairs. In this paper, we propose a novel third bypass called clan embedding. Here each point x is mapped to a subset of points f(x), called a clan, with a special chief point χ(x)∈ f(x). The clan embedding has multiplicative distortion t if for every pair (x,y) some copy y′∈ f(y) in the clan of y is close to the chief of x: miny′∈ f(y)d(y′,χ(x))≤ t· d(x,y). Our first result is a clan embedding into a tree with multiplicative distortion O(logn/є) such that each point has 1+є copies (in expectation). In addition, we provide a “spanning” version of this theorem for graphs and use it to devise the first compact routing scheme with constant size routing tables.

The FOCS'19 paper of Le and Solomon [Hung Le and Shay Solomon, 2019], culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy (1+ε)-spanner in ℝ^d is Õ(ε^{-d}) for any d = O(1) and any ε = Ω(n^{-1/(d-1)}) (where Õ hides polylogarithmic factors of 1/ε), and also shows the existence of point sets in ℝ^d for which any (1+ε)-spanner must have lightness Ω(ε^{-d}). Given this tight bound on the lightness, a natural arising question is whether a better lightness bound can be achieved using Steiner points. Our first result is a construction of Steiner spanners in ℝ² with lightness O(ε^{-1} log Δ), where Δ is the spread of the point set. In the regime of Δ ≪ 2^(1/ε), this provides an improvement over the lightness bound of [Hung Le and Shay Solomon, 2019]; this regime of parameters is of practical interest, as point sets arising in real-life applications (e.g., for various random distributions) have polynomially bounded spread, while in spanner applications ε often controls the precision, and it sometimes needs to be much smaller than O(1/log n). Moreover, for spread polynomially bounded in 1/ε, this upper bound provides a quadratic improvement over the non-Steiner bound of [Hung Le and Shay Solomon, 2019], We then demonstrate that such a light spanner can be constructed in O_ε(n) time for polynomially bounded spread, where O_ε hides a factor of poly(1/(ε)). Finally, we extend the construction to higher dimensions, proving a lightness upper bound of Õ(ε^{-(d+1)/2} + ε^{-2} log Δ) for any 3 ≤ d = O(1) and any ε = Ω(n^{-1/(d-1)}).

Understanding the structure of minor-free metrics, namely shortest path metrics obtained over a weighted graph excluding a fixed minor, has been an important research direction since the fundamental work of Robertson and Seymour. A fundamental idea that helps both to understand the structural properties of these metrics and lead to strong algorithmic results is to construct a “small-complexity” graph that approximately preserves distances between pairs of points of the metric. We show the two following structural results for minor-free metrics: 1) Construction of a light subset spanner. Given a subset of vertices called terminals, and ε, in polynomial time we construct a sub graph that preserves all pairwise distances between terminals up to a multiplicative 1+ε factor, of total weight at most Oε(1) times the weight of the minimal Steiner tree spanning the terminals. 2) Construction of a stochastic metric embedding into low treewidth graphs with expected additive distortion εD. Namely, given a minor-free graph G = (V, E, w) of diameter D, and parameter ε, we construct a distribution D over dominating metric embeddings into treewidth- Oε(logn) graphs such that ∀u, v ∈ V, \mathbbEf ~ D[dH(f(u), f(v))] ≤ dG(u, v)+εD. Our results have the following algorithmic consequences: (1) the first efficient approximation scheme for subset TSP in minor-free metrics; (2) the first approximation scheme for bounded-capacity vehicle routing in minor-free metrics; (3) the first efficient approximation scheme for bounded-capacity vehicle routing on bounded genus metrics. En route to the latter result, we design the first FPT approximation scheme for bounded-capacity vehicle routing on bounded-treewidth graphs (parameterized by the treewidth).

We give the first PTAS for the subset Traveling Salesperson Problem (TSP) in H-minor-free graphs. This resolves a long standing open problem in a long line of work on designing PTASes for TSP in minor-closed families initiated by Grigni, Koutsoupias and Papadimitriou in FOCS’95. The main technical ingredient in our PTAS is a construction of a nearly light subset (1 + ϵ)-spanner for any given edge-weighted H-minor-free graph. This construction is based on a necessary and sufficient condition given by sparse spanner oracles: light subset spanners exist if and only if sparse spanner oracles exist. This relationship allows us to obtain two new results: An (1 + ε)-spanner with lightness O(ϵ−d+2) for any doubling metric of constant dimension d. This improves the earlier lightness bound ϵ−O(d) obtained by Borradaile, Le and Wulff-Nilsen [14]. An (1 + ϵ)-spanner with sublinear lightness for any metric of constant correlation dimension. Previously, no spanner with non-trivial lightness was known.

Euclidean spanners are important geometric structures, having found numerous applications over the years. Cornerstone results in this area from the late 80s and early 90s state that for any d-dimensional n-point Euclidean space, there exists a (1+ε) -spanner with O(nε^-d+1) edges and lightness (normalized weight) O(ε^-2d)^1. Surprisingly, the fundamental question of whether or not these dependencies on ε and d for small d can be improved has remained elusive, even for d = 2. This question naturally arises in any application of Euclidean spanners where precision is a necessity (thus ε is tiny). In the most extreme case ε is inverse polynomial in n, and then one could potentially improve the size and lightness bounds by factors that are polynomial in n. The state-of-the-art bounds O(nε^-d+1) and O(ε^-2d) on the size and lightness of spanners are realized by the greedy spanner. In 2016, Filtser and Solomon [25] proved that, in low dimensional spaces, the greedy spanner is “near-optimal''; informally, their result states that the greedy spanner for dimension d is just as sparse and light as any other spanner but for dimension larger by a constant factor. Hence the question of whether the greedy spanner is truly optimal remained open to date. The contribution of this paper is two-fold. 1) We resolve these longstanding questions by nailing down the exact dependencies on ε and d and showing that the greedy spanner is truly optimal. Specifically, for any d= O(1), ε = Ω(n^-1/d-1): • We show that any (1+ε) -spanner must have Ω(nε^-d+1) edges, implying that the greedy (and other) spanners achieve the optimal size. • We show that any (1+ε) -spanner must have lightness Ω(ε^-d), and then improve the upper bound on the lightness of the greedy spanner from O(ε^-2d) to Õ_ε (ε^-d). 2) We then complement our negative result for the size of spanners with a rather counterintuitive positive result: Steiner points lead to a quadratic improvement in the size of spanners! Our bound for the size of Steiner spanners is tight as well (up to lower-order terms).

Lightness and sparsity are two natural parameters for Euclidean $(1+\varepsilon)$-spanners. Classical results show that, when the dimension $d\in \mathbb{N}$ and $\varepsilon>0$ are constant, every set $S$ of $n$ points in $d$-space admits a $(1+\varepsilon)$-spanner with $O(n)$ edges and weight proportional to that of the Euclidean minimum spanning tree of $S$. In a recent breakthrough, Le and Solomon [Proceedings of FOCS, 2019, pp. 1078--1100] established the precise dependencies on $\varepsilon>0$, for constant $d\in \mathbb{N}$, of the minimum lightness and sparsity of $(1+\varepsilon)$-spanners, and observed that Steiner points can substantially improve the lightness and sparsity of a $(1+\varepsilon)$-spanner. They gave upper bounds of $\tilde{O}(\varepsilon^{-(d+1)/2})$ for the minimum lightness in dimensions $d\geq 3$ and $\tilde{O}(\varepsilon^{-(d-1)/2})$ for the minimum sparsity in $d$-space for all $d\geq 1$. Subsequently, Le and Solomon [LIPIcs Leibniz Int. Proc. Inform. 173, Schloss Dagstuhl, Wadern, 2020, pp. 67:1--67:22] constructed Steiner $(1+\varepsilon)$-spanners of lightness $O(\varepsilon^{-1}\log\Delta)$ in the plane, where $\Delta\in \Omega(\sqrt{n})$ is the spread of $S$, defined as the ratio between the maximum and the minimum distance between a pair of points. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner $(1+\varepsilon)$-spanners. We establish lower bounds of $\Omega(\varepsilon^{-d/2})$ for the lightness and $\Omega(\varepsilon^{-(d-1)/2})$ for the sparsity of such spanners in Euclidean $d$-space for all constant $d\geq 2$. Our lower bound constructions generalize previous constructions by Le and Solomon, but the analysis substantially simplifies previous work, using new geometric insight, focusing on the directions of edges. Next, we show that for every finite set of points in the plane and every $\varepsilon\in (0,1]$, there exists a Euclidean Steiner $(1+\varepsilon)$-spanner of lightness $O(\varepsilon^{-1})$; this matches the lower bound for $d=2$. We generalize the notion of shallow light trees, which may be of independent interest, and use directional spanners and a modified window partitioning scheme to achieve a tight weight analysis.

We show that every H-minor-free graph has a light (1+≥ilon)-spanner, resolving an open problem of Grigni and Sissokho and proving a conjecture of Grigni and Hung \cite{GH12}. Our lightness bound is \[O\left(\frac{\sigma_H}{≥ilon^3}\log \frac{1}{≥ilon}\right)\] where \sigma_H = |V(H)|√{\log |V(H)|} is the sparsity coefficient of H-minor-free graphs. That is, it has a practical dependency on the size of the minor H. Our result also implies that the polynomial time approximation scheme (PTAS) for the Travelling Salesperson Problem (TSP) in H-minor-free graphs by Demaine, Hajiaghayi and Kawarabayashi is an efficient PTAS whose running time is 2^{O_H\left(\frac{1}{≥ilon^4}\log \frac{1}{≥ilon}\right)}n^{O(1)} where O_H ignores dependencies on the size of H. Our techniques significantly deviate from existing lines of research on spanners for H-minor-free graphs, but build upon the work of Chechik and Wulff-Nilsen for spanners of general graphs[6].

There has been recent progress in showing that the exponential dependence on treewidth in dynamic programming algorithms for solving NP-hard problems is optimal under the Strong Exponential Time Hypothesis (SETH). We extend this work to r-domination problems. In r-dominating set, one wishes to find a minimum subset S of vertices such that every vertex of G is within r hops of some vertex in S. In connected r-dominating set, one additionally requires that the set induces a connected subgraph of G. We give a O((2r+1)^tw n) time algorithm for r-dominating set and a randomized O((2r+2)^tw n^{O(1)}) time algorithm for connected r-dominating set in n-vertex graphs of treewidth tw. We show that the running time dependence on r and tw is the best possible under SETH. This adds to earlier observations that a "+1" in the denominator is required for connectivity constraints.