arXiv subjects

Sándor Kisfaludi-Bak

Publications and source records attributed to Sándor Kisfaludi-Bak.

At least 19 recordsLinked to original sources

arXiv2026

Metric Planarity Testing is Hard

A key milestone in algorithmic graph theory is Hopcroft and Tarjan's 1974 linear-time algorithm for graph planarity testing. At a workshop in 2002, Linial and Rabinovich asked for either a polynomial-time algorithm or an NP-hardness proof for metric planarity testing: given a metric space $(X,d_X)$, determine whether there exists an edge-weighted planar graph $G=(V,E,w)$ with $X\subseteq V$ that realizes the distances in $X$, meaning that $d_G(x,y)=d_X(x,y)$ for every $x,y\in X$. This question arose in the context of research on metric embeddings and their applications in computer science since the mid-1990s. Over the past few decades, polynomial-time algorithms have been developed for several special cases. In this work, we show that metric planarity testing is NP-hard, in sharp contrast to graph planarity testing. Our reduction yields a stronger result: under the Exponential Time Hypothesis (ETH), no $2^{o(n)}$-time algorithm for metric planarity testing exists, where $n$ is the number of points in the metric. To complement our ETH lower bound, we give two algorithmic results: (i) an algorithm for computing a metric planar embedding in $2^{\tilde{O}(n^2)}$ time and (ii) a $(1+ε)$-approximation algorithm in $2^{\tilde{O}(n)}$ time for any fixed $ε\in(0,1)$. The second algorithm produces a planar graph that preserves the input metric up to a factor of $(1+ε)$.

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arXiv2026

Fast Thick-Thin Decomposition for Sparse Spanners on Hyperbolic Surfaces

We consider spanners for point sets lying in the hyperbolic plane or on a closed hyperbolic surface with the restriction that spanner edges are not allowed to cross. This is a natural generalization of non-crossing Euclidean spanners. Thus, the resulting spanner graphs are embedded in the hyperbolic plane or on the hyperbolic surface. As our main contribution, we show that there are sparse $(1+\varepsilon)$-spanners for these problems when we are allowed to use Steiner points: - on the hyperbolic plane we get a non-crossing Steiner $(1+\varepsilon)$-spanner with $\mathcal{O}(n / \varepsilon^2)$ edges, - on hyperbolic surfaces of genus $g$ we get a Steiner $(1+\varepsilon)$-spanner with $\mathcal{O}(n / \varepsilon^{3/2} + g/\varepsilon^2)$ non-crossing edges, or with $\mathcal{O}(n / \sqrt{\varepsilon} + g/\varepsilon)$ edges that are allowed to cross. In particular, our spanners on surfaces have sparsity with linear dependence on $g$, rather than the easier-to-attain exponential dependence, and the terms $n/\varepsilon^{3/2}$ and $n/\sqrt{\varepsilon}$ match the current best Euclidean results for plane and crossing Steiner spanners, respectively. As a corollary of our non-crossing spanner and techniques from the existing literature on light spanners and minor-free TSP, we get an EPTAS for TSP on hyperbolic surfaces. Our surface constructions rely on the thick-thin decomposition, a standard tool for studying hyperbolic surfaces. For convex hyperbolic polygons, we introduce an analogous neck decomposition. We give algorithms that compute the thick-thin decomposition of a genus-$g$ surface in $\mathcal{O}(g^4\log g)$ time and the neck decomposition of an $n$-vertex polygon in $\mathcal{O}(n)$ time.

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arXiv2026

Shifting is Optimal under Gap-ETH: A Lower Bound Framework for Geometric Approximation Schemes

The shifting technique of Hochbaum and Maass [J.ACM'85] produces PTASes with the fastest known running times $n^{O(1/\varepsilon^{d-1})}$ for several $d$-dimensional geometric problems. However, it is only known, due to Marx [FOCS'07], that these algorithms are indeed optimal for dimension $d=2$. We show that these running times are optimal under Gap-ETH for every constant dimension. More precisely, we develop a framework that enables us to prove the conditional optimality of the shifting algorithms for several problems on unit ball graphs, such as maximum independent set, maximum induced forest, and others, as well as for the problem of piercing unit balls. Our framework is built using the cube wiring theorem of De Berg et al. [SICOMP'20] and the reduction steps of Marx and Sidiropoulos [SoCG'14] to create a convenient maximization version of geometric CSP that can be used as a basis for reductions.

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arXiv2026

Touring a Sequence of Orthogonal Polygons

We study the problem of computing a shortest tour that visits a sequence of $k$ polygons $P_1,\dots, P_k$ with a total number of $n$ vertices. A tour is an oriented curve such that there exist points $p_i\in P_i$ for all $i$ where $p_i$ appears not after $p_{i+1}$. In a seminal paper, Dror, Efrat, Lubiw and Mitchell (STOC 2003) considered the problem under $L_2$ distance, and gave $\widetilde O(nk)$ and $\widetilde O(nk^2)$ algorithms for disjoint and intersecting convex polygons, respectively. In this paper, we consider the orthogonal setting (with orthogonal polygons and Manhattan distance) and obtain the following results: - a truly subquadratic $\widetilde O(n^{2-\frac{1}{48}})$ algorithm when consecutive polygons in the sequence are disjoint; - an $\widetilde O(n)$ algorithm for ortho-convex polygons when consecutive polygons are disjoint; - an $O(n)$ algorithm for axis-aligned rectangles; - $\widetilde O(n^2)$ and $\widetilde O(n^{1.5}k^2)$ algorithms without restrictions. Our algorithms build on a wide range of techniques, including additively weighted Voronoi diagrams, rectangle decompositions, persistent data structures, and dynamic distance oracles for weighted planar graphs.

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arXiv2026

Charting the Diameter Computation Landscape on Intersection Graphs in the Plane

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, 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. 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 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 rely on succinct representation assuming bounded VC-dimension; for such a strategy $Ω(n^{7/4})$ time is an inherent barrier. 3. An $\tilde{O}(n^{4/3})$-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 $\tilde{O}(n^{2-1/9})$. 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 only hold when deciding if diameter is at most 3.

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arXiv2026

Approximation Schemes for Subset TSP and Steiner Tree on Geometric Intersection Graphs

We give approximation schemes for Subset TSP and Steiner Tree on unit disk graphs, and more generally, on intersection graphs of similarly sized connected fat (not necessarily convex) polygons in the plane. As a first step towards this goal, we prove spanner-type results: finding an induced subgraph of bounded size that is $(1+\varepsilon)$-equivalent to the original instance in the sense that the optimum value increases only by a factor of at most $(1+\varepsilon)$ when the solution can use only the edges in this subgraph. - For Subset TSP, our algorithms find a $(1+\varepsilon)$-equivalent induced subgraph of size $\mathrm{poly}(1/\varepsilon)\cdot\mathrm{OPT}$ in polynomial time, and use it to find a $(1+\varepsilon)$-approximate solution in time $2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$. - For Steiner Tree, our algorithms find a $(1+\varepsilon)$-equivalent induced subgraph of size $2^{\mathrm{poly}(1/\varepsilon)}\cdot\mathrm{OPT}$ in time $2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$, and use it to find a $(1+\varepsilon)$-approximate solution in time $2^{2^{\mathrm{poly}(1/\varepsilon)}}\cdot n^{O(1)}$. - An improved algorithm finds a $(1+\varepsilon)$-approximate solution for Steiner Tree in time $2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$. An easy reduction shows that approximation schemes for unit disks imply approximation schemes for planar graphs. Thus our results are far-reaching generalizations of analogous results of Klein [STOC'06] and Borradaile, Klein, and Mathieu [ACM TALG'09] for Subset TSP and Steiner Tree in planar graphs. We show that our results are best possible in the sense that dropping any of (i) similarly sized, (ii) connected, or (iii) fat makes both problems APX-hard.

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arXiv2026

Charting the Diameter Computation Landscape of Geometric Intersection Graphs in Three Dimensions and Higher

Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case 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. [SoCG '22] 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: - 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. - 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 truly subquadratic hard when the diameter is $Ω(\log n)$. - A near-linear-time algorithm for Diameter-2 of unit cubes in 3D, generalizing the previous result for unit squares in 2D. - A truly subquadratic-time algorithm and lower bound for Diameter-2 and Diameter-3 of rectangular boxes (of arbitrary dimension and sizes), respectively.

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arXiv2026

Gap-ETH-Tight Algorithms for Hyperbolic TSP and Steiner Tree

We give an approximation scheme for the TSP in $d$-dimensional hyperbolic space that has optimal dependence on $\varepsilon$ under Gap-ETH. For any fixed dimension $d\geq 2$ and for any $\varepsilon>0$ our randomized algorithm gives a $(1+\varepsilon)$-approximation in time $2^{O(1/\varepsilon^{d-1})}n^{1+o(1)}$. We also provide an algorithm for the hyperbolic Steiner tree problem with the same running time. Our algorithm is an Arora-style dynamic program based on a randomly shifted hierarchical decomposition. However, we introduce a new hierarchical decomposition called the hybrid hyperbolic quadtree to achieve the desired large-scale structure, which deviates significantly from the recently proposed hyperbolic quadtree of Kisfaludi-Bak and Van Wordragen (JoCG'25). Moreover, we have a new non-uniform portal placement, and our structure theorem employs a new weighted crossing analysis. We believe that these techniques could form the basis for further developments in geometric optimization in curved spaces.

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arXiv2026

Recognizing Subgraphs of Regular Tilings

For $p,q\ge2$ the $\{p,q\}$-tiling graph is the (finite or infinite) planar graph $T_{p,q}$ where all faces are cycles of length $p$ and all vertices have degree $q$. We give algorithms for the problem of recognizing (induced) subgraphs of these graphs, as follows. - For $1/p+1/q>1/2$, these graphs correspond to regular tilings of the sphere. These graphs are finite, thus recognizing their (induced) subgraphs can be done in constant time. - For $1/p+1/q=1/2$, these graphs correspond to regular tilings of the Euclidean plane. For the Euclidean square grid $T_{4,4}$ Bhatt and Cosmadakis (IPL'87) showed that recognizing subgraphs is NP-hard, even if the input graph is a tree. We show that a simple divide-and conquer algorithm achieves a subexponential running time in all Euclidean tilings, and we observe that there is an almost matching lower bound in $T_{4,4}$ under the Exponential Time Hypothesis via known reductions. - For $1/p+1/q<1/2$, these graphs correspond to regular tilings of the hyperbolic plane. As our main contribution, we show that deciding if an $n$-vertex graph is isomorphic to a subgraph of the tiling $T_{p,q}$ can be done in quasi-polynomial ($n^{O(\log n)}$) time for any fixed $q$. Our results for the hyperbolic case show that it has significantly lower complexity than the Euclidean variant, and it is unlikely to be NP-hard. The Euclidean results also suggest that the problem can be maximally hard even if the graph in question is a tree. Consequently, the known treewidth bounds for subgraphs of hyperbolic tilings do not lead to an efficient algorithm by themselves. Instead, we use convex hulls within the tiling graph, which have several desirable properties in hyperbolic tilings. Our key technical insight is that planar subgraph isomorphism can be computed via a dynamic program that builds a sphere cut decomposition of a solution subgraph's convex hull.

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arXiv2026

Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity

A Euclidean noncrossing Steiner $(1+ε)$-spanner for a point set $P\subset\mathbb{R}^2$ is a planar straight-line graph that, for any two points $a, b \in P$, contains a path whose length is at most $1+ε$ times the Euclidean distance between $a$ and $b$. We construct a Euclidean noncrossing Steiner $(1+ε)$-spanner with $O(n/ε^{3/2})$ edges for any set of $n$ points in the plane. This result improves upon the previous best upper bound of $O(n/ε^{4})$ obtained nearly three decades ago. We also establish an almost matching lower bound: There exist $n$ points in the plane for which any Euclidean noncrossing Steiner $(1+ε)$-spanner has $Ω_μ(n/ε^{3/2-μ})$ edges for any $μ>0$. Our lower bound uses recent generalizations of the Szemerédi-Trotter theorem to disk-tube incidences in geometric measure theory.

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arXiv2026

Lower Bounds for Dominating Set in Ball Graphs and for Weighted Dominating Set in Unit-Ball Graphs

Recently it was shown that many classic graph problems -- Independent Set, Dominating Set, Hamiltonian Cycle, and more -- can be solved in subexponential time on unit-ball graphs. More precisely, these problems can be solved in $2^{O(n^{1-1/d})}$ time on unit-ball graphs in $\mathbb R^d$, which is tight under ETH. The result can be generalized to intersection graphs of similarly-sized fat objects. For Independent Set the same running time can be achieved for non-similarly-sized fat objects, and for the weighted version of the problem. We show that such generalizations most likely are not possible for Dominating Set: assuming ETH, we prove that - there is no algorithm with running time $2^{o(n)}$ for Dominating Set on (non-unit) ball graphs in $\mathbb R^3$; - there is no algorithm with running time $2^{o(n)}$ for Weighted Dominating Set on unit-ball graphs in $\mathbb R^3$; - there is no algorithm with running time $2^{o(n)}$ for Dominating Set, Connected Dominating Set, or Steiner Tree on intersections graphs of arbitrary convex (but non-constant-complexity) objects in the plane.

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arXiv2025

Shortest Paths, Convexity, and Treewidth in Regular Hyperbolic Tilings

Hyperbolic tilings are natural infinite planar graphs where each vertex has degree $q$ and each face has $p$ edges for some $\frac1p+\frac1q<\frac12$. We study the structure of shortest paths in such graphs. We show that given a set of $n$ terminals, we can compute a so-called isometric closure (closely related to the geodesic convex hull) of the terminals in near-linear time, using a classic geometric convex hull algorithm as a black box. We show that the size of the convex hull is $O(N)$ where $N$ is the total length of the paths to the terminals from a fixed origin. Furthermore, we prove that the geodesic convex hull of a set of $n$ terminals has treewidth only $\max(12,O(\log\frac{n}{p + q}))$, a bound independent of the distance of the points involved. As a consequence, we obtain algorithms for subset TSP and Steiner tree with running time $O(N \log N) + \mathrm{poly}(\frac{n}{p + q}) \cdot N$.

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arXiv2025

Truly Subquadratic Time Algorithms for Diameter and Related Problems in Graphs of Bounded VC-dimension

We give the first truly subquadratic time algorithm, with $O^*(n^{2-1/18})$ 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 $\tilde{O}(mn^{1-1/(2d)})$ 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 Potȩpa [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 $\tilde{O}(n^{2-1/12})$ time algorithm for computing the diameter of intersection graphs of axis-aligned squares with arbitrary size. The best-known algorithm by Duraj, Konieczny, and Potȩpa [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. (... truncated to meet the arXiv abstract requirement.)

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arXiv2025

Realizing Metric Spaces with Convex Obstacles

The presence of obstacles has a major impact on distance computation, motion planning, and visibility. While these problems are well studied in the plane, our understanding in three and higher dimensions is still limited. We investigate how different obstacle properties affect the induced geodesic metric in three-dimensional space. A finite metric space is said to be approximately realizable by a collection of obstacles if, for any $\varepsilon>0$, it can be embedded into the free space around the obstacles with geodesic distance and worst-case distortion $1+\varepsilon$. We focus on three key properties-convexity, disjointness, and fatness-and analyze how omitting each of them influences realizability. Our main result shows that if fatness is dropped, then every finite metric space can be realized with distortion $1+\varepsilon$ using convex, pairwise disjoint obstacles in $\mathbb{R}^3$, even if all obstacles are congruent equilateral triangles. Moreover, if we enforce fatness but drop convexity or disjointness, the same realizability still holds. Our results have important implications on the approximability of TSP with Obstacles, a natural variant of TSP introduced recently by Alkema et al. (ESA 2022). Specifically, we use the recent results of Banerjee et al. on TSP in doubling spaces (FOCS 2024) and of Chew et al. on distances among obstacles (Information Processing Letters 2002) to show that TSP with Obstacles admits a PTAS if the obstacles are convex, fat, and pairwise disjoint. If any of these three properties is dropped, then our results, combined with the APX-hardness of Metric TSP, demonstrate that TSP with Obstacles is APX-hard.

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arXiv2025

Near-Optimal Dynamic Steiner Spanners for Constant-Curvature Spaces

We consider Steiner spanners in Euclidean and non-Euclidean geometries. In the Euclidean setting, a recent line of work initiated by Le and Solomon [FOCS'19] and further improved by Chang et al. [SoCG'24] obtained Steiner $(1+\varepsilon)$-spanners of size $O_d(\varepsilon^{(1-d)/2}\log(1/\varepsilon)n)$, nearly matching the lower bounds of Bhore and Tóth [SIDMA'22]. We obtain Steiner $(1+\varepsilon)$-spanners of size $O_d(\varepsilon^{(1-d)/2}\log(1/\varepsilon)n)$ not only in $d$-dimensional Euclidean space, but also in $d$-dimensional spherical and hyperbolic space. For any fixed dimension $d$, the obtained edge count is optimal up to an $O(\log(1/\varepsilon))$ factor in each of these spaces. Unlike earlier constructions, our Steiner spanners are based on simple quadtrees, and they can be dynamically maintained, leading to efficient data structures for dynamic approximate nearest neighbours and bichromatic closest pair. In the hyperbolic setting, we also show that $2$-spanners in the hyperbolic plane must have $Ω(n\log n)$ edges, and we obtain a $2$-spanner of size $O_d(n\log n)$ in $d$-dimensional hyperbolic space, matching our lower bound for any constant $d$. Finally, we give a Steiner spanner with additive error $\varepsilon$ in hyperbolic space with $O_d(\varepsilon^{(1-d)/2}\log(α(n)/\varepsilon)n)$ edges, where $α(n)$ is the inverse Ackermann function. Our techniques generalize to closed orientable surfaces of constant curvature as well as to some quotient spaces.

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arXiv2025

Structure and Independence in Hyperbolic Uniform Disk Graphs

We consider intersection graphs of disks of radius $r$ in the hyperbolic plane. Unlike the Euclidean setting, these graph classes are different for different values of $r$, where very small $r$ corresponds to an almost-Euclidean setting and $r \in Ω(\log n)$ corresponds to a firmly hyperbolic setting. We observe that larger values of $r$ create simpler graph classes, at least in terms of separators and the computational complexity of the \textsc{Independent Set} problem. First, we show that intersection graphs of disks of radius $r$ in the hyperbolic plane can be separated with $\mathcal{O}((1+1/r)\log n)$ cliques in a balanced manner. Our second structural insight concerns Delaunay complexes in the hyperbolic plane and may be of independent interest. We show that for any set $S$ of $n$ points with pairwise distance at least $2r$ in the hyperbolic plane the corresponding Delaunay complex has outerplanarity $1+\mathcal{O}(\frac{\log n}{r})$, which implies a similar bound on the balanced separators and treewidth of such Delaunay complexes. Using this outerplanarity (and treewidth) bound we prove that \textsc{Independent Set} can be solved in $n^{\mathcal{O}(1+\frac{\log n}{r})}$ time. The algorithm is based on dynamic programming on some unknown sphere cut decomposition that is based on the solution. The resulting algorithm is a far-reaching generalization of a result of Kisfaludi-Bak (SODA 2020), and it is tight under the Exponential Time Hypothesis. In particular, \textsc{Independent Set} is polynomial-time solvable in the firmly hyperbolic setting of $r\in Ω(\log n)$. Finally, in the case when the disks have ply (depth) at most $\ell$, we give a PTAS for \textsc{Maximum Independent Set} that has only quasi-polynomial dependence on $1/\varepsilon$ and $\ell$. Our PTAS is a further generalization of our exact algorithm.

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arXiv2024

A Gap-ETH-Tight Approximation Scheme for Euclidean TSP

We revisit the classic task of finding the shortest tour of $n$ points in $d$-dimensional Euclidean space, for any fixed constant $d \geq 2$. We determine the optimal dependence on $\varepsilon$ in the running time of an algorithm that computes a $(1+\varepsilon)$-approximate tour, under a plausible assumption. Specifically, we give an algorithm that runs in $2^{\mathcal{O}(1/\varepsilon^{d-1})} n\log n$ time. This improves the previously smallest dependence on $\varepsilon$ in the running time $(1/\varepsilon)^{\mathcal{O}(1/\varepsilon^{d-1})}n \log n$ of the algorithm by Rao and Smith~(STOC 1998). We also show that a $2^{o(1/\varepsilon^{d-1})}\text{poly}(n)$ algorithm would violate the Gap-Exponential Time Hypothesis (Gap-ETH). Our new algorithm builds upon the celebrated quadtree-based methods initially proposed by Arora (J. ACM 1998), but it adds a new idea that we call \emph{sparsity-sensitive patching}. On a high level this lets the granularity with which we simplify the tour depend on how sparse it is locally. We demonstrate that our technique extends to other problems, by showing that for Steiner Tree and Rectilinear Steiner Tree it yields the same running time. We complement our results with a matching Gap-ETH lower bound for Rectilinear Steiner Tree.

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arXiv2024

Euclidean TSP in Narrow Strips

We investigate how the complexity of Euclidean TSP for point sets $P$ inside the strip $(-\infty,+\infty)\times [0,δ]$ depends on the strip width $δ$. We obtain two main results. First, for the case where the points have distinct integer $x$-coordinates, we prove that a shortest bitonic tour (which can be computed in $O(n\log^2 n)$ time using an existing algorithm) is guaranteed to be a shortest tour overall when $δ\leq 2\sqrt{2}$, a bound which is best possible. Second, we present an algorithm that is fixed-parameter tractable with respect to $δ$. Our algorithm has running time $2^{O(\sqrtδ)} n + O(δ^2 n^2)$ for sparse point sets, where each $1\timesδ$ rectangle inside the strip contains $O(1)$ points. For random point sets, where the points are chosen uniformly at random from the rectangle $[0,n]\times [0,δ]$, it has an expected running time of $2^{O(\sqrtδ)} n$. These results generalise to point sets $P$ inside a hypercylinder of width $δ$. In this case, the factors $2^{O(\sqrtδ)}$ become $2^{O(δ^{1-1/d})}$.

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