arXiv subjects

Linda Kleist

Publications and source records attributed to Linda Kleist.

At least 19 recordsLinked to original sources

arXiv2026

Online Geometric Packing through Online TSP Scheduling

We consider the problem of online packing of convex polygons into a strip by translations. While online algorithms with a constant competitive ratio have been known for rectangles for decades [Baker and Schwarz, SICOMP 1983], the current best algorithm for convex polygons has competitive ratio $O(n^{\log_2 3-1}\log n) = O(n^{0.59})$, where $n$ is the number of polygons. This algorithm was described by Aamand, Abrahamsen, Beretta, and Kleist [SODA 2023], who also proved a lower bound of $Ω(\sqrt{\log n/\log\log n})$ on the competitive ratio of any algorithm. Their lower bound is obtained via a reduction from \emph{online sorting}, a problem introduced in the same paper, for which they established a lower bound on the competitive ratio. We introduce a new, natural online problem that we call online TSP scheduling. Here, points $x_1,\ldots,x_n$ arrive online from a metric space $(M,d)$, and upon arrival each $x_i$ must be assigned a visit time $p_i\in[0,\infty)$ satisfying $|p_i-p_j|\ge d(x_i,x_j)$ for all $j<i$. The cost of the schedule is $\max_i p_i$. We present an $O(\log^2 n)$-competitive algorithm for online TSP scheduling, and show how this implies an $O(\log^2 n)$-competitive algorithm for online translational strip packing of convex polygons. We also prove that the same competitive ratio is achievable for other translational packing problems, including online packing of $d$-dimensional unit hyperdisks in $\mathbb R^{d+1}$, whose offline version was studied by Alt, Cabello, Cheong, Park, and Seiferth [Comp. Geom. 2026]. Our algorithm for online TSP scheduling builds on a recent breakthrough for online sorting by Azar, Panigrahi, and Vardi [SODA 2026]. We thus show that the connection between packing and online sorting can be used not only for lower bounds, but also for algorithms.

cs.CG↗

arXiv2026

Hamilton paths and cycles in flip graphs of (almost-)perfect matchings

We consider the set of matchings of a graph and a local change operation, called a flip, between them. In the combinatorial setting, the base graphs are either complete graphs or complete bipartite graphs, and in the geometric setting, the graphs are embedded on point sets in the plane, with the requirement that edges must be drawn as straight lines and must not cross. For base graphs with an even number of vertices, we consider perfect matchings, i.e., all vertices are matched, and for base graphs with an odd number of vertices, we consider almost-perfect matchings, i.e., all but one vertex of the graph are matched. A 2-flip between two perfect matchings exchanges two edges, and a 1-flip between two almost-perfect matchings exchanges one edge. The corresponding flip graph has the set of perfect or almost-perfect matchings as vertices, with pairs of them connected by an edge if they differ in a 2-flip or 1-flip, respectively. In this work, we provide a comprehensive picture of Hamiltonicity properties of these flip graphs. We prove that the flip graphs in the combinatorial setting are Hamilton-connected, i.e., they admit a Hamilton path between any two vertices, or, if the flip graphs are bipartite, we prove that they are Hamilton-laceable, i.e., they admit a Hamilton path between any two vertices from different partition classes. In the geometric setting, we prove that any path in them misses exponentially many vertices, in particular, they have no Hamilton paths or cycles. For points in convex position and almost-perfect matchings under 1-flips, we complement this by constructing a cycle in the flip graph that visits almost all vertices.

math.CO↗

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.

cs.CG↗

arXiv2026

Covering and Partitioning Complex Objects with Small Pieces

We study the problems of covering or partitioning a polygon $P$ (possibly with holes) using a minimum number of small pieces, where a small piece is a connected sub-polygon contained in an axis-aligned unit square. For covering, we seek to write $P$ as a union of small pieces, and in partitioning, we furthermore require the pieces to be pairwise interior-disjoint. We show that these problems are in fact equivalent: Optimum covers and partitions have the same number of pieces. For covering, a natural local search algorithm repeatedly attempts to replace $k$ pieces from a candidate cover with $k-1$ pieces. In two dimensions and for sufficiently large $k$, we show that when no such swap is possible, the cover is a $1+O(1/\sqrt k)$-approximation, hence obtaining the first PTAS for the problem. Prior to our work, the only known algorithm was a $13$-approximation that only works for polygons without holes [Abrahamsen and Rasmussen, SODA 2025]. In contrast, in the three dimensional version of the problem, for a polyhedron $P$ of complexity $n$, we show that it is NP-hard to approximate an optimal cover or partition to within a factor that is logarithmic in $n$, even if $P$ is simple, i.e., has genus $0$ and no holes.

cs.CG↗

arXiv2026

Online Packing of Orthogonal Polygons

While rectangular and box-shaped objects dominate the classic discourse of theoretic investigations, a fascinating frontier lies in packing more complex shapes. Given recent insights that convex polygons do not allow for constant competitive online algorithms for diverse variants under translation, we study orthogonal polygons, in particular of small complexity. For translational packings of orthogonal 6-gons, we show that the competitive ratio of any online algorithm that aims to pack the items into a minimal number of unit bins is in $Ω(n / \log n)$, where $n$ denotes the number of objects. In contrast, we show that constant competitive algorithms exist when the orthogonal 6-gons are symmetric or small. For (orthogonally convex) orthogonal 8-gons, we show that the trivial $n$-competitive algorithm, which places each item in its own bin, is best-possible, i.e., every online algorithm has an asymptotic competitive ratio of at least $n$. This implies that for general orthogonal polygons, the trivial algorithm is best possible. Interestingly, for packing degenerate orthogonal polygons (with thickness $0$), called skeletons, the change in complexity is even more drastic. While constant competitive algorithms for 6-skeletons exist, no online algorithm for 8-skeletons achieves a competitive ratio better than $n$. For other packing variants of orthogonal 6-gons under translation, our insights imply the following consequences. The asymptotic competitive ratio of any online algorithm is in $Ω(n / \log n)$ for strip packing, and there exist online algorithms with competitive ratios in $O(1)$ for perimeter packing, or in $O(\sqrt{n})$ for minimizing the area of the bounding box. Moreover, the critical packing density is positive (if every object individually fits into the interior of a unit bin).

cs.CG↗

arXiv2026

Flip Distance of Non-Crossing Spanning Trees: NP-Hardness and Improved Bounds

We consider the problem of reconfiguring non-crossing spanning trees on point sets. For a set $P$ of $n$ points in general position in the plane, the flip graph $F(P)$ has a vertex for each non-crossing spanning tree on $P$ and an edge between any two spanning trees that can be transformed into each other by the exchange of a single edge. This flip graph has been intensively studied, lately with an emphasis on determining its diameter diam$(F(P))$ for sets $P$ of $n$ points in convex position. The current best bounds are $\frac{14}{9}n-O(1) \leq$ diam$(F(P))<\frac{15}{9}n-3$ [Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber; SODA 2025]. The crucial tool for both the upper and lower bound are so-called *conflict graphs*, which the authors stated might be the key ingredient for determining the diameter (up to lower-order terms). In this paper, we pick up the concept of conflict graphs and show that this tool is even more versatile than previously hoped. As our first main result, we use conflict graphs to show that computing the flip distance between two non-crossing spanning trees is NP-hard, even for point sets in convex position. Interestingly, the result still holds for more constrained flip operations, concretely, compatible flips (where the removed and the added edge do not cross) and rotations (where the removed and the added edge share an endpoint). Extending the line of research from [BKUV SODA25], we present new insights on the diameter of the flip graph. Their lower bound is based on a constant-size pair of trees, one of which is *stacked*. We show that if one of the trees is stacked, then the lower bound is indeed optimal up to a constant term, that is, there exists a flip sequence of length at most $\frac{14}{9}(n-1)$ to any other tree. Lastly, we improve the lower bound on the diameter of the flip graph $F(P)$ for $n$ points in convex position to $\frac{11}{7}n-o(n)$.

cs.CG↗

arXiv2026

Disproving two conjectures on the Hamiltonicity of Venn diagrams

In 1984, Winkler conjectured that every simple Venn diagram with $n$ curves can be extended to a simple Venn diagram with $n+1$ curves. His conjecture is equivalent to the statement that the dual graph of any simple Venn diagram has a Hamilton cycle. In this work, we construct counterexamples to Winkler's conjecture for all $n\geq 6$. As part of this proof, we computed all 3.430.404 simple Venn diagrams with $n=6$ curves (even their number was not previously known), among which we found 72 counterexamples. We also construct monotone Venn diagrams, i.e., diagrams that can be drawn with $n$ convex curves, and are not extendable, for all $n\geq 7$. Furthermore, we also disprove another conjecture about the Hamiltonicity of the (primal) graph of a Venn diagram. Specifically, while working on Winkler's conjecture, Pruesse and Ruskey proved that this graph has a Hamilton cycle for every simple Venn diagram with $n$ curves, and conjectured that this also holds for non-simple diagrams. We construct counterexamples to this conjecture for all $n\geq 4$.

math.CO↗

arXiv2025

The Price of Connectivity Augmentation on Planar Graphs

Given two classes of graphs, $\mathcal{G}_1\subseteq \mathcal{G}_2$, and a $c$-connected graph $G\in \mathcal{G}_1$, we wish to augment $G$ with a smallest cardinality set of new edges $F$ to obtain a $k$-connected graph $G'=(V,E\cup F) \in \mathcal{G}_2$. In general, this is the $c\to k$ connectivity augmentation problem. Previous research considered variants where $\mathcal{G}_1=\mathcal{G}_2$ is the class of planar graphs, plane graphs, or planar straight-line graphs. In all three settings, we prove that the $c\to k$ augmentation problem is NP-complete when $2\leq c<k\leq 5$. However, the connectivity of the augmented graph $G'$ is at most $5$ if $\mathcal{G}_2$ is limited to planar graphs. We initiate the study of the $c\to k$ connectivity augmentation problem for arbitrary $k\in \mathbb{N}$, where $\mathcal{G}_1$ is the class of planar graphs, plane graphs, or planar straight-line graphs, and $\mathcal{G}_2$ is a beyond-planar class of graphs: $\ell$-planar, $\ell$-plane topological, or $\ell$-plane geometric graphs. We obtain tight bounds on the tradeoffs between the desired connectivity $k$ and the local crossing number $\ell$ of the augmented graph $G'$. We also show that our hardness results apply to this setting. The connectivity augmentation problem for triangulations is intimately related to edge flips; and the minimum augmentation problem to the flip distance between triangulations. We prove that it is NP-complete to find the minimum flip distance between a given triangulation and a 4-connected triangulation, settling an open problem posed in 2014, and present an EPTAS for this problem.

cs.CG↗

arXiv2025

An Improved Bound for Plane Covering Paths

A covering path for a finite set $P$ of points in the plane is a polygonal path such that every point of $P$ lies on a segment of the path. The vertices of the path need not be at points of $P$. A covering path is plane if its segments do not cross each other. Let $π(n)$ be the minimum number such that every set of $n$ points in the plane admits a plane covering path with at most $π(n)$ segments. We prove that $π(n)\le \lceil6n/7\rceil$. This improves the previous best-known upper bound of $\lceil 21n/22\rceil$, due to Biniaz (SoCG 2023). Our proof is constructive and yields a simple $O(n\log n)$-time algorithm for computing a plane covering path.

cs.CG↗

arXiv2025

Using Mobile Relays to Strongly Connect a Minimum-Power Network between Terminals Complying with No-Transmission Zones

We present strategies for placing a swarm of mobile relays to provide a bi-directional wireless network that connects fixed (immobile) terminals. Neither terminals nor relays are permitted to transmit into disk-shaped no-transmission zones. We assume a planar environment and that each transmission area is a disk centered at the transmitter. We seek a strongly connected network between all terminals with minimal total cost, where the cost is the sum area of the transmission disks. Results for networks with increasing levels of complexity are provided. The solutions for local networks containing low numbers of relays and terminals are applied to larger networks. For more complex networks, algorithms for a minimum-spanning tree (MST) based procedure are implemented to reduce the solution cost. A procedure to characterize and determine the possible homotopies of a system of terminals and obstacles is described, and used to initialize the evolution of the network under the presented algorithms.

cs.NI↗

arXiv2025

Augmenting Plane Straight-Line Graphs to Meet Parity Constraints

Given a plane geometric graph $G$ on $n$ vertices, we want to augment it so that given parity constraints of the vertex degrees are met. In other words, given a subset $R$ of the vertices, we are interested in a plane geometric supergraph $G'$ such that exactly the vertices of $R$ have odd degree in $G'\setminus G$. We show that the question whether such a supergraph exists can be decided in polynomial time for two interesting cases. First, when the vertices are in convex position, we present a linear-time algorithm. Building on this insight, we solve the case when $G$ is a plane geometric path in $O(n \log n)$ time. This solves an open problem posed by Catana, Olaverri, Tejel, and Urrutia (Appl. Math. Comput. 2020).

cs.CG↗

arXiv2024

Reconfiguration of unit squares and disks: PSPACE-hardness in simple settings

We study two well-known reconfiguration problems. Given a start and a target configuration of geometric objects in a polygon, we wonder whether we can move the objects from the start configuration to the target configuration while avoiding collisions between the objects and staying within the polygon. Problems of this type have been considered since the early 80s by roboticists and computational geometers. In this paper, we study some of the simplest possible variants where the objects are unlabeled unit squares or unit disks. In unlabeled reconfiguration, the objects are identical, so that any object is allowed to end at any of the targets positions. We show that it is PSPACE-hard to decide whether there exists a reconfiguration of unit squares even in a simple polygon. Previously, it was only known to be PSPACE-hard in a polygon with holes [Solovey and Halperin, Int. J. Robotics Res. 2016]. Our proof is based on a result of independent interest, namely that reconfiguration between two satisfying assignments of a formula of Monotone-Planar-3SAT is also PSPACE-complete. The reduction from reconfiguration of Monotone-Planar-3SAT to reconfiguration of unit squares extends techniques recently developed to show NP-hardness of packing unit squares in a simple polygon [Abrahamsen and Stade, FOCS 2024]. We also show PSPACE-hardness of reconfiguration of unit disks in a polygon with holes. Previously, it was only known that reconfiguration of disks of two different sizes was PSPACE-hard [Brocken, van der Heijden, Kostitsyna, Lo-Wong and Surtel, FUN 2021].

cs.CG↗

arXiv2024

Facet-Hamiltonicity

We consider facet-Hamiltonian cycles of polytopes, defined as cycles in their skeleton such that every facet is visited exactly once. These cycles can be understood as optimal watchman routes that guard the facets of a polytope. We consider the existence of such cycles for a variety of polytopes, the facets of which have a natural combinatorial interpretation. In particular, we prove the following results: - Every permutahedron has a facet-Hamiltonian cycle. These cycles consist of circular sequences of permutations of $n$ elements, where two successive permutations differ by a single adjacent transposition, and such that every subset of $[n]$ appears as a prefix in a contiguous subsequence. With these cycles we associate what we call rhombic strips which encode interleaved Gray codes of the Boolean lattice, one Gray code for each rank. These rhombic strips correspond to simple Venn diagrams. - Every generalized associahedron has a facet-Hamiltonian cycle. This generalizes the so-called rainbow cycles of Felsner, Kleist, Mütze, and Sering (SIDMA 2020) to associahedra of any finite type. We relate the constructions to the Conway-Coxeter friezes and the bipartite belts of finite type cluster algebras. - Graph associahedra of wheels, fans, and complete split graphs have facet-Hamiltonian cycles. For associahedra of complete bipartite graphs and caterpillars, we construct facet-Hamiltonian paths. The construction involves new insights on the combinatorics of graph tubings. We also consider the computational complexity of deciding whether a given polytope has a facet-Hamiltonian cycle and show that the problem is NP-complete, even when restricted to simple 3-dimensional polytopes.

math.CO↗

arXiv2024

Flipping Non-Crossing Spanning Trees

For a set $P$ of $n$ points in general position in the plane, the flip graph $F(P)$ has a vertex for each non-crossing spanning tree on $P$ and an edge between any two spanning trees that can be transformed into each other by one edge flip. The diameter ${\rm diam}(F(P))$ of this graph is subject of intensive study. For points in general position, it is between $3n/2-5$ and $2n-4$, with no improvement for 25 years. For points in convex position, it lies between $3n/2 - 5$ and $\approx1.95n$, where the lower bound was conjectured to be tight up to an additive constant and the upper bound is a recent breakthrough improvement over several bounds of the form $2n-o(n)$. In this work, we provide new upper and lower bounds on ${\rm diam}(F(P))$, mainly focusing on points in convex position. We show $14n/9 - O(1) \le {\rm diam}(F(P)) \le 5n/3 - 3$, by this disproving the conjectured upper bound of $3n/2$ for convex position, and relevantly improving both the long-standing lower bound for general position and the recent new upper bound for convex position. We complement these by showing that if one of $T,T'$ has at most two boundary edges, then ${\rm dist}(T,T') \le 2d/2 < 3n/2$, where $d = |T-T'|$ is the number of edges in one tree that are not in the other. To prove both the upper and the lower bound, we introduce a new powerful tool. Specifically, we convert the flip distance problem for given $T,T'$ to the problem of a largest acyclic subset in an associated conflict graph $H(T,T')$. In fact, this method is powerful enough to give an equivalent formulation of the diameter of $F(P)$ for points $P$ in convex position up to lower-order terms. As such, conflict graphs are likely the key to a complete resolution of this and possibly also other reconfiguration problems.

cs.CG↗

arXiv2024

Minimum Plane Bichromatic Spanning Trees

For a set of red and blue points in the plane, a minimum bichromatic spanning tree (MinBST) is a shortest spanning tree of the points such that every edge has a red and a blue endpoint. A MinBST can be computed in $O(n\log n)$ time where $n$ is the number of points. In contrast to the standard Euclidean MST, which is always plane (noncrossing), a MinBST may have edges that cross each other. However, we prove that a MinBST is quasi-plane, that is, it does not contain three pairwise crossing edges, and we determine the maximum number of crossings. Moreover, we study the problem of finding a minimum plane bichromatic spanning tree (MinPBST) which is a shortest bichromatic spanning tree with pairwise noncrossing edges. This problem is known to be NP-hard. The previous best approximation algorithm, due to Borgelt et al. (2009), has a ratio of $O(\sqrt{n})$. It is also known that the optimum solution can be computed in polynomial time in some special cases, for instance, when the points are in convex position, collinear, semi-collinear, or when one color class has constant size. We present an $O(\log n)$-factor approximation algorithm for the general case.

cs.CG↗

arXiv2024

Targeted Drug Delivery: Algorithmic Methods for Collecting a Swarm of Particles with Uniform External Forces

We investigate algorithmic approaches for targeted drug delivery in a complex, maze-like environment, such as a vascular system. The basic scenario is given by a large swarm of micro-scale particles (''agents'') and a particular target region (''tumor'') within a system of passageways. Agents are too small to contain on-board power or computation and are instead controlled by a global external force that acts uniformly on all particles, such as an applied fluidic flow or electromagnetic field. The challenge is to deliver all agents to the target region with a minimum number of actuation steps. We provide a number of results for this challenge. We show that the underlying problem is NP-complete, which explains why previous work did not provide provably efficient algorithms. We also develop several algorithmic approaches that greatly improve the worst-case guarantees for the number of required actuation steps. We evaluate our algorithmic approaches by numerous simulations, both for deterministic algorithms and searches supported by deep learning, which show that the performance is practically promising.

cs.CG↗

arXiv2024

On the Connectivity of the Flip Graph of Plane Spanning Paths

Flip graphs of non-crossing configurations in the plane are widely studied objects, e.g., flip graph of triangulations, spanning trees, Hamiltonian cycles, and perfect matchings. Typically, it is an easy exercise to prove connectivity of a flip graph. In stark contrast, the connectivity of the flip graph of plane spanning paths on point sets in general position has been an open problem for more than 16 years. In order to provide new insights, we investigate certain induced subgraphs. Firstly, we provide tight bounds on the diameter and the radius of the flip graph of spanning paths on points in convex position with one fixed endpoint. Secondly, we show that so-called suffix-independent paths induce a connected subgraph. Consequently, to answer the open problem affirmatively, it suffices to show that each path can be flipped to some suffix-independent path. Lastly, we investigate paths where one endpoint is fixed and provide tools to flip to suffix-independent paths. We show that these tools are strong enough to show connectivity of the flip graph of plane spanning paths on point sets with at most two convex layers.

cs.CG↗

arXiv2024

Online Sorting and Translational Packing of Convex Polygons

We investigate several online packing problems in which convex polygons arrive one by one and have to be placed irrevocably into a container, while the aim is to minimize the used space. Among other variants, we consider strip packing and bin packing, where the container is the infinite horizontal strip $[0,\infty)\times [0,1]$ or a collection of $1 \times 1$ bins, respectively. We draw interesting connections to the following online sorting problem OnlineSorting$[γ,n]$: We receive a stream of real numbers $s_1,\ldots,s_n$, $s_i\in[0,1]$, one by one. Each real must be placed in an array $A$ with $γn$ initially empty cells without knowing the subsequent reals. The goal is to minimize the sum of differences of consecutive reals in $A$. The offline optimum is to place the reals in sorted order so the cost is at most $1$. We show that for any $Δ$-competitive online algorithm of OnlineSorting$[γ,n]$, it holds that $γΔ\inΩ(\log n/\log \log n)$. We use this lower bound to prove the non-existence of competitive algorithms for various online translational packing problems of convex polygons, among them strip packing, bin packing and perimeter packing. This also implies that there exists no online algorithm that can pack all streams of pieces of diameter and total area at most $δ$ into the unit square. These results are in contrast to the case when the pieces are restricted to rectangles, for which competitive algorithms are known. Likewise, the offline versions of packing convex polygons have constant factor approximation algorithms. As a complement, we also include algorithms for both online sorting and translation-only online strip packing with non-trivial competitive ratios. Our algorithm for strip packing relies on a new technique for recursively subdividing the strip into parallelograms of varying height, thickness and slope.

cs.CG↗