arXiv subjects

Wolfgang Mulzer

Publications and source records attributed to Wolfgang Mulzer.

At least 19 recordsLinked to original sources

arXiv2025

Near-Linear and Parameterized Approximations for Maximum Cliques in Disk Graphs

A \emph{disk graph} is the intersection graph of (closed) disks in the plane. We consider the classic problem of finding a maximum clique in a disk graph. For general disk graphs, the complexity of this problem is still open, but for unit disk graphs, it is well known to be in P. The currently fastest algorithm runs in time $O(n^{7/3+ o(1)})$, where $n$ denotes the number of disks~\cite{EspenantKM23, keil_et_al:LIPIcs.SoCG.2025.63}. Moreover, for the case of disk graphs with $t$ distinct radii, the problem has also recently been shown to be in XP. More specifically, it is solvable in time $O^*(n^{2t})$~\cite{keil_et_al:LIPIcs.SoCG.2025.63}. In this paper, we present algorithms with improved running times by allowing for approximate solutions and by using randomization: - for unit disk graphs, we give an algorithm that, with constant success probability, computes a $(1-\varepsilon)$-approximate maximum clique in expected time $\tilde{O}(n/\varepsilon^2)$; and - for disk graphs with $t$ distinct radii, we give a parameterized approximation scheme that, with a constant success probability, computes a $(1-\varepsilon)$-approximate maximum clique in expected time $\tilde{O}(f(t)\cdot (1/\varepsilon)^{O(t)} \cdot n)$, for some (exponential) function $f(t)$.

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arXiv2024

Robust Algorithms for Finding Triangles and Computing the Girth in Unit Disk and Transmission Graphs

We describe optimal robust algorithms for finding a triangle and the unweighted girth in a unit disk graph, as well as finding a triangle in a transmission graph.In the robust setting, the input is not given as a set of sites in the plane, but rather as an abstract graph. The input may or may not be realizable as a unit disk graph or a transmission graph. If the graph is realizable, the algorithm is guaranteed to give the correct answer. If not, the algorithm will either give a correct answer or correctly state that the input is not of the required type.

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arXiv2023

Insertion-Only Dynamic Connectivity in General Disk Graphs

Let $S \subseteq \mathbb{R}^2$ be a set of $n$ \emph{sites} in the plane, so that every site $s \in S$ has an \emph{associated radius} $r_s > 0$. Let $D(S)$ be the \emph{disk intersection graph} defined by $S$, i.e., the graph with vertex set $S$ and an edge between two distinct sites $s, t \in S$ if and only if the disks with centers $s$, $t$ and radii $r_s$, $r_t$ intersect. Our goal is to design data structures that maintain the connectivity structure of $D(S)$ as $S$ changes dynamically over time. We consider the incremental case, where new sites can be inserted into $S$. While previous work focuses on data structures whose running time depends on the ratio between the smallest and the largest site in $S$, we present a data structure with $O(\alpha(n))$ amortized query time and $O(\log^6 n)$ expected amortized insertion time.

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arXiv2022

Well-Separation and Hyperplane Transversals in High Dimensions

A family of $k$ point sets in $d$ dimensions is well-separated if the convex hulls of any two disjoint subfamilies can be separated by a hyperplane. Well-separation is a strong assumption that allows us to conclude that certain kinds of generalized ham-sandwich cuts for the point sets exist. But how hard is it to check if a given family of high-dimensional point sets has this property? Starting from this question, we study several algorithmic aspects of the existence of transversals and separations in high-dimensions. First, we give an explicit proof that $k$ point sets are well-separated if and only if their convex hulls admit no $(k - 2)$-transversal, i.e., if there exists no $(k - 2)$-dimensional flat that intersects the convex hulls of all $k$ sets. It follows that the task of checking well-separation lies in the complexity class coNP. Next, we show that it is NP-hard to decide whether there is a hyperplane-transversal (that is, a $(d - 1)$-transversal) of a family of $d + 1$ line segments in $\mathbb{R}^d$, where $d$ is part of the input. As a consequence, it follows that the general problem of testing well-separation is coNP-complete. Furthermore, we show that finding a hyperplane that maximizes the number of intersected sets is NP-hard, but allows for an $\Omega\left(\frac{\log k}{k \log \log k}\right)$-approximation algorithm that is polynomial in $d$ and $k$, when each set consists of a single point. When all point sets are finite, we show that checking whether there exists a $(k - 2)$-transversal is in fact strongly NP-complete.

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arXiv2022

Unique Sink Orientations of Grids is in Unique End of Potential Line

The complexity classes Unique End of Potential Line (UEOPL) and its promise version PUEOPL were introduced in 2018 by Fearnly et al. UEOPL captures search problems where the instances are promised to have a unique solution. UEOPL captures total search versions of these promise problems. The promise problems can be made total by defining violations that are returned as a short certificate of an unfulfilled promise. GridUSO is the problem of finding the sink in a grid with a unique sink orientation. It was introduced by G\"artner et al. We describe a promise preserving reduction from GridUSO to UniqueForwardEOPL, a UEOPL-complete problem. Thus, we show that GridUSO is in UEOPL and its promise version is in PUEOPL.

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arXiv2022

Nearest-Neighbor Decompositions of Drawings

Let $\mathcal{D}$ be a set of straight-line segments in the plane, potentially crossing, and let $c$ be a positive integer. We denote by $P$ the union of the endpoints of the straight-line segments of $\mathcal{D}$ and of the intersection points between pairs of segments. We say that $\mathcal{D}$ has a nearest-neighbor decomposition into $c$ parts if we can partition $P$ into $c$ point sets $P_1, \ldots, P_c$ such that $\mathcal{D}$ is the union of the nearest neighbor graphs on $P_1, \ldots, P_c$. We show that it is NP-complete to decide whether $\mathcal{D}$ can be drawn as the union of $c\geq 3$ nearest-neighbor graphs, even when no two segments cross. We show that for $c = 2$, it is NP-complete in the general setting and polynomial-time solvable when no two segments cross. We show the existence of an $O(\log n)$-approximation algorithm running in subexponential time for partitioning $\mathcal{D}$ into a minimum number of nearest-neighbor graphs. As a main tool in our analysis, we establish the notion of the conflict graph for a drawing $\mathcal{D}$. The vertices of the conflict graph are the connected components of $\mathcal{D}$, with the assumption that each connected component is the nearest neighbor graph of its vertices, and there is an edge between two components $U$ and $V$ if and only if the nearest neighbor graph of $U \cup V$ contains an edge between a vertex in $U$ and a vertex in $V$. We show that string graphs are conflict graphs of certain planar drawings. For planar graphs and complete $k$-partite graphs, we give additional, more efficient constructions. We furthermore show that there are subdivisions of non-planar graphs that are not conflict graphs. Lastly, we show a separator lemma for conflict graphs.

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arXiv2022

Compatible Spanning Trees in Simple Drawings of $K_n$

For a simple drawing $D$ of the complete graph $K_n$, two (plane) subdrawings are compatible if their union is plane. Let $\mathcal{T}_D$ be the set of all plane spanning trees on $D$ and $\mathcal{F}(\mathcal{T}_D)$ be the compatibility graph that has a vertex for each element in $\mathcal{T}_D$ and two vertices are adjacent if and only if the corresponding trees are compatible. We show, on the one hand, that $\mathcal{F}(\mathcal{T}_D)$ is connected if $D$ is a cylindrical, monotone, or strongly c-monotone drawing. On the other hand, we show that the subgraph of $\mathcal{F}(\mathcal{T}_D)$ induced by stars, double stars, and twin stars is also connected. In all cases the diameter of the corresponding compatibility graph is at most linear in $n$.

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arXiv2022

Flipping Plane Spanning Paths

Let $S$ be a planar point set in general position, and let $\mathcal{P}(S)$ be the set of all plane straight-line paths with vertex set $S$. A flip on a path $P \in \mathcal{P}(S)$ is the operation of replacing an edge $e$ of $P$ with another edge $f$ on $S$ to obtain a new valid path from $\mathcal{P}(S)$. It is a long-standing open question whether for every given point set $S$, every path from $\mathcal{P}(S)$ can be transformed into any other path from $\mathcal{P}(S)$ by a sequence of flips. To achieve a better understanding of this question, we show that it is sufficient to prove the statement for plane spanning paths whose first edge is fixed. Furthermore, we provide positive answers for special classes of point sets, namely, for wheel sets and generalized double circles (which include, e.g., double chains and double circles).

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arXiv2021

Dynamic Connectivity in Disk Graphs

Let $S$ be a set of $n$ sites in the plane, so that every site $s \in S$ has an associated radius $r_s > 0$. Let $\mathcal{D}(S)$ be the disk intersection graph defined by $S$, i.e., the graph with vertex set $S$ and an edge between two distinct sites $s, t \in S$ if and only if the disks with centers $s$, $t$ and radii $r_s$, $r_t$ intersect.Our goal is to design data structures that maintain the connectivity structure of $\mathcal{D}(S)$ as sites are inserted and/or deleted in $S$.

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arXiv2021

Long Plane Trees

In the longest plane spanning tree problem, we are given a finite planar point set $\mathcal{P}$, and our task is to find a plane (i.e., noncrossing) spanning tree for $\mathcal{P}$ with maximum total Euclidean edge length. Despite more than two decades of research, it remains open whether this problem is NP-hard. Thus, previous efforts have focused on olynomial-time algorithms that produce plane trees whose total edge length approximates $\text{OPT}$, the maximum possible length. The approximate trees in these algorithms all have small unweighted diameter, typically three or four. It is natural to ask whether this is a common feature of longest plane spanning trees, or an artifact of the specific approximation algorithms. We provide three results to elucidate the interplay between the approximation guarantee and the unweighted diameter of the approximate trees. First, we describe a polynomial-time algorithm to construct a plane tree with diameter at most four and total edge length at least $0.546 \cdot \text{OPT}$. This constitutes a substantial improvement over the state of the art. Second, we show that a longest plane tree among those with diameter at most three can be found in polynomial time. Third, for any candidate diameter $d \geq 3$, we provide upper bounds on the approximation factor that can be achieved by a longest plane tree with diameter at most $d$ (compared to a longest plane tree without constraints).

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arXiv2020

Minimum Cuts in Geometric Intersection Graphs

Let $\mathcal{D}$ be a set of $n$ disks in the plane. The disk graph $G_\mathcal{D}$ for $\mathcal{D}$ is the undirected graph with vertex set $\mathcal{D}$ in which two disks are joined by an edge if and only if they intersect. The directed transmission graph $G^{\rightarrow}_\mathcal{D}$ for $\mathcal{D}$ is the directed graph with vertex set $\mathcal{D}$ in which there is an edge from a disk $D_1 \in \mathcal{D}$ to a disk $D_2 \in \mathcal{D}$ if and only if $D_1$ contains the center of $D_2$. Given $\mathcal{D}$ and two non-intersecting disks $s, t \in \mathcal{D}$, we show that a minimum $s$-$t$ vertex cut in $G_\mathcal{D}$ or in $G^{\rightarrow}_\mathcal{D}$ can be found in $O(n^{3/2}\text{polylog} n)$ expected time. To obtain our result, we combine an algorithm for the maximum flow problem in general graphs with dynamic geometric data structures to manipulate the disks. As an application, we consider the barrier resilience problem in a rectangular domain. In this problem, we have a vertical strip $S$ bounded by two vertical lines, $L_\ell$ and $L_r$, and a collection $\mathcal{D}$ of disks. Let $a$ be a point in $S$ above all disks of $\mathcal{D}$, and let $b$ a point in $S$ below all disks of $\mathcal{D}$. The task is to find a curve from $a$ to $b$ that lies in $S$ and that intersects as few disks of $\mathcal{D}$ as possible. Using our improved algorithm for minimum cuts in disk graphs, we can solve the barrier resilience problem in $O(n^{3/2}\text{polylog} n)$ expected time.

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arXiv2020

A Simple Randomized $O(n \log n)$--Time Closest-Pair Algorithm in Doubling Metrics

Consider a metric space $(P,dist)$ with $N$ points whose doubling dimension is a constant. We present a simple, randomized, and recursive algorithm that computes, in $O(N \log N)$ expected time, the closest-pair distance in $P$. To generate recursive calls, we use previous results of Har-Peled and Mendel, and Abam and Har-Peled for computing a sparse annulus that separates the points in a balanced way.

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arXiv2020

Long Alternating Paths Exist

Let $P$ be a set of $2n$ points in convex position, such that $n$ points are colored red and $n$ points are colored blue. A non-crossing alternating path on $P$ of length $\ell$ is a sequence $p_1, \dots, p_\ell$ of $\ell$ points from $P$ so that (i) all points are pairwise distinct; (ii) any two consecutive points $p_i$, $p_{i+1}$ have different colors; and (iii) any two segments $p_i p_{i+1}$ and $p_j p_{j+1}$ have disjoint relative interiors, for $i \neq j$. We show that there is an absolute constant $\varepsilon > 0$, independent of $n$ and of the coloring, such that $P$ always admits a non-crossing alternating path of length at least $(1 + \varepsilon)n$. The result is obtained through a slightly stronger statement: there always exists a non-crossing bichromatic separated matching on at least $(1 + \varepsilon)n$ points of $P$. This is a properly colored matching whose segments are pairwise disjoint and intersected by common line. For both versions, this is the first improvement of the easily obtained lower bound of $n$ by an additive term linear in $n$. The best known published upper bounds are asymptotically of order $4n/3+o(n)$.

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arXiv2020

Computational Complexity of the $\alpha$-Ham-Sandwich Problem

The classic Ham-Sandwich theorem states that for any $d$ measurable sets in $\mathbb{R}^d$, there is a hyperplane that bisects them simultaneously. An extension by B\'ar\'any, Hubard, and Jer\'onimo [DCG 2008] states that if the sets are convex and \emph{well-separated}, then for any given $\alpha_1, \dots, \alpha_d \in [0, 1]$, there is a unique oriented hyperplane that cuts off a respective fraction $\alpha_1, \dots, \alpha_d$ from each set. Steiger and Zhao [DCG 2010] proved a discrete analogue of this theorem, which we call the \emph{$\alpha$-Ham-Sandwich theorem}. They gave an algorithm to find the hyperplane in time $O(n (\log n)^{d-3})$, where $n$ is the total number of input points. The computational complexity of this search problem in high dimensions is open, quite unlike the complexity of the Ham-Sandwich problem, which is now known to be PPA-complete (Filos-Ratsikas and Goldberg [STOC 2019]). Recently, Fearley, Gordon, Mehta, and Savani [ICALP 2019] introduced a new sub-class of CLS (Continuous Local Search) called \emph{Unique End-of-Potential Line} (UEOPL). This class captures problems in CLS that have unique solutions. We show that for the $\alpha$-Ham-Sandwich theorem, the search problem of finding the dividing hyperplane lies in UEOPL. This gives the first non-trivial containment of the problem in a complexity class and places it in the company of classic search problems such as finding the fixed point of a contraction map, the unique sink orientation problem and the $P$-matrix linear complementarity problem.

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arXiv2020

Routing in Unit Disk Graphs without Dynamic Headers

Let $V\subset\mathbb{R}^2$ be a set of $n$ sites in the plane. The unit disk graph $DG(V)$ of $V$ is the graph with vertex set $V$ in which two sites $v$ and $w$ are adjacent if and only if their Euclidean distance is at most $1$. We develop a compact routing scheme for $DG(V)$. The routing scheme preprocesses $DG(V)$ by assigning a label $l(v)$ to every site $v$ in $V$. After that, for any two sites $s$ and $t$, the scheme must be able to route a packet from $s$ to $t$ as follows: given the label of a current vertex $r$ (initially, $r=s$) and the label of the target vertex $t$, the scheme determines a neighbor $r'$ of $r$. Then, the packet is forwarded to $r'$, and the process continues until the packet reaches its desired target $t$. The resulting path between the source $s$ and the target $t$ is called the routing path of $s$ and $t$. The stretch of the routing scheme is the maximum ratio of the total Euclidean length of the routing path and of the shortest path in $DG(V)$, between any two sites $s, t \in V$. We show that for any given $\varepsilon>0$, we can construct a routing scheme for $DG(V)$ with diameter $D$ that achieves stretch $1+\varepsilon$ and label size $O(\log D\log^3n/\log\log n)$ (the constant in the $O$-Notation depends on $\varepsilon$). In the past, several routing schemes for unit disk graphs have been proposed. Our scheme is the first one to achieve poly-logarithmic label size and arbitrarily small stretch without storing any additional information in the packet.

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arXiv2020

The Tree Stabbing Number is not Monotone

Let $P \subseteq \mathbb{R}^2$ be a set of points and $T$ be a spanning tree of $P$. The \emph{stabbing number} of $T$ is the maximum number of intersections any line in the plane determines with the edges of $T$. The \emph{tree stabbing number} of $P$ is the minimum stabbing number of any spanning tree of $P$. We prove that the tree stabbing number is not a monotone parameter, i.e., there exist point sets $P \subsetneq P'$ such that \treestab{$P$} $>$ \treestab{$P'$}, answering a question by Eppstein \cite[Open Problem~17.5]{eppstein_2018}.

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arXiv2019

Maximum Matchings in Geometric Intersection Graphs

Let $G$ be an intersection graph of $n$ geometric objects in the plane. We show that a maximum matching in $G$ can be found in $O(\rho^{3\omega/2}n^{\omega/2})$ time with high probability, where $\rho$ is the density of the geometric objects and $\omega>2$ is a constant such that $n \times n$ matrices can be multiplied in $O(n^\omega)$ time. The same result holds for any subgraph of $G$, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators. We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in $O(n^{\omega/2})$ time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in $[1, \Psi]$ can be found in $O(\Psi^6\log^{11} n + \Psi^{12 \omega} n^{\omega/2})$ time with high probability.

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arXiv2019

Minimal Representations of Order Types by Geometric Graphs

In order to have a compact visualization of the order type of a given point set S, we are interested in geometric graphs on S with few edges that unambiguously display the order type of S. We introduce the concept of exit edges, which prevent the order type from changing under continuous motion of vertices. That is, in the geometric graph on S whose edges are the exit edges, in order to change the order type of S, at least one vertex needs to move across an exit edge. Exit edges have a natural dual characterization, which allows us to efficiently compute them and to bound their number.

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