arXiv subjects

Martijn Struijs

Publications and source records attributed to Martijn Struijs.

6 recordsLinked to original sources

arXiv2026

Optimal Motion Planning for Two Square Robots in a Rectilinear Environment

Let $\mathcal{W} \subset \mathbb{R}^2$ be a rectilinear polygonal environment (that is, a rectilinear polygon potentially with holes) with a total of $n$ vertices, and let $A,B$ be two robots, each modeled as an axis-aligned unit square, that can move rectilinearly inside $\mathcal{W}$. The goal is to compute a collision-free motion plan $\boldsymbolπ$, that is, a motion plan that continuously moves $A$ from $s_A$ to $t_A$ and $B$ from $s_B$ to $t_B$ so that $A$ and $B$ remain inside $\mathcal{W}$ and do not collide with each other during the motion. We study two variants of this problem which are focused additionally on the optimality of $\boldsymbolπ$, and obtain the following results. 1. Min-Sum: Here the goal is to compute a motion plan that minimizes the sum of the lengths of the paths of the robots. We present an $O(n^4\log{n})$-time algorithm for computing an optimal solution to the min-sum problem. This is the first polynomial-time algorithm to compute an optimal, collision-free motion of two robots amid obstacles in a planar polygonal environment. 2. Min-Makespan: Here the robots can move with at most unit speed, and the goal is to compute a motion plan that minimizes the maximum time taken by a robot to reach its target location. We prove that the min-makespan variant is NP-hard.

cs.CG↗

arXiv2022

Near-Shortest Path Routing in Hybrid Communication Networks

Hybrid networks, i.e., networks that leverage different means of communication, become ever more widespread. To allow theoretical study of such networks, [Augustine et al., SODA'20] introduced the $\mathsf{HYBRID}$ model, which is based on the concept of synchronous message passing and uses two fundamentally different principles of communication: a local mode, which allows every node to exchange one message per round with each neighbor in a local communication graph; and a global mode where any pair of nodes can exchange messages, but only few such exchanges can take place per round. A sizable portion of the previous research for the $\mathsf{HYBRID}$ model revolves around basic communication primitives and computing distances or shortest paths in networks. In this paper, we extend this study to a related fundamental problem of computing compact routing schemes for near-shortest paths in the local communication graph. We demonstrate that, for the case where the local communication graph is a unit-disc graph with $n$ nodes that is realized in the plane and has no radio holes, we can deterministically compute a routing scheme that has constant stretch and uses labels and local routing tables of size $O(\log n)$ bits in only $O(\log n)$ rounds.

cs.DC↗

arXiv2021

Minimum Scan Cover and Variants -- Theory and Experiments

We consider a spectrum of geometric optimization problems motivated by contexts such as satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs, we are given a graph $G$ that is embedded in Euclidean space. The edges of $G$ need to be scanned, i.e., probed from both of their vertices. In order to scan their edge, two vertices need to face each other; changing the heading of a vertex incurs some cost in terms of energy or rotation time that is proportional to the corresponding rotation angle. Our goal is to compute schedules that minimize the following objective functions: (i) in Minimum Makespan Scan Cover (MSC-MS), this is the time until all edges are scanned; (ii) in Minimum Total Energy Scan Cover (MSC-TE), the sum of all rotation angles; (iii) in Minimum Bottleneck Energy Scan Cover (MSC-BE), the maximum total rotation angle at one vertex. Previous theoretical work on MSC-MS revealed a close connection to graph coloring and the cut cover problem, leading to hardness and approximability results. In this paper, we present polynomial-time algorithms for 1D instances of MSC-TE and MSC-BE, but NP-hardness proofs for bipartite 2D instances. For bipartite graphs in 2D, we also give 2-approximation algorithms for both MSC-TE and MSC-BE. Most importantly, we provide a comprehensive study of practical methods for all three problems. We compare three different mixed-integer programming and two constraint programming approaches, and show how to compute provably optimal solutions for geometric instances with up to 300 edges. Additionally, we compare the performance of different meta-heuristics for even larger instances.

cs.CG↗

arXiv2020

On the hardness of computing an average curve

We study the complexity of clustering curves under $k$-median and $k$-center objectives in the metric space of the Fréchet distance and related distance measures. Building upon recent hardness results for the minimum-enclosing-ball problem under the Fréchet distance, we show that also the $1$-median problem is NP-hard. Furthermore, we show that the $1$-median problem is W[1]-hard with the number of curves as parameter. We show this under the discrete and continuous Fréchet and Dynamic Time Warping (DTW) distance. This yields an independent proof of an earlier result by Bulteau et al. from 2018 for a variant of DTW that uses squared distances, where the new proof is both simpler and more general. On the positive side, we give approximation algorithms for problem variants where the center curve may have complexity at most $\ell$ under the discrete Fréchet distance. In particular, for fixed $k,\ell$ and $\varepsilon$, we give $(1+\varepsilon)$-approximation algorithms for the $(k,\ell)$-median and $(k,\ell)$-center objectives and a polynomial-time exact algorithm for the $(k,\ell)$-center objective.

cs.CG↗

arXiv2019

Fast Distributed Algorithms for LP-Type Problems of Bounded Dimension

In this paper we present various distributed algorithms for LP-type problems in the well-known gossip model. LP-type problems include many important classes of problems such as (integer) linear programming, geometric problems like smallest enclosing ball and polytope distance, and set problems like hitting set and set cover. In the gossip model, a node can only push information to or pull information from nodes chosen uniformly at random. Protocols for the gossip model are usually very practical due to their fast convergence, their simplicity, and their stability under stress and disruptions. Our algorithms are very efficient (logarithmic rounds or better with just polylogarithmic communication work per node per round) whenever the combinatorial dimension of the given LP-type problem is constant, even if the size of the given LP-type problem is polynomially large in the number of nodes.

cs.DC↗

arXiv2018

Approximating $(k,\ell)$-center clustering for curves

The Euclidean $k$-center problem is a classical problem that has been extensively studied in computer science. Given a set $\mathcal{G}$ of $n$ points in Euclidean space, the problem is to determine a set $\mathcal{C}$ of $k$ centers (not necessarily part of $\mathcal{G}$) such that the maximum distance between a point in $\mathcal{G}$ and its nearest neighbor in $\mathcal{C}$ is minimized. In this paper we study the corresponding $(k,\ell)$-center problem for polygonal curves under the Fréchet distance, that is, given a set $\mathcal{G}$ of $n$ polygonal curves in $\mathbb{R}^d$, each of complexity $m$, determine a set $\mathcal{C}$ of $k$ polygonal curves in $\mathbb{R}^d$, each of complexity $\ell$, such that the maximum Fréchet distance of a curve in $\mathcal{G}$ to its closest curve in $\mathcal{C}$ is minimized. In this paper, we substantially extend and improve the known approximation bounds for curves in dimension $2$ and higher. We show that, if $\ell$ is part of the input, then there is no polynomial-time approximation scheme unless $\mathsf{P}=\mathsf{NP}$. Our constructions yield different bounds for one and two-dimensional curves and the discrete and continuous Fréchet distance. In the case of the discrete Fréchet distance on two-dimensional curves, we show hardness of approximation within a factor close to $2.598$. This result also holds when $k=1$, and the $\mathsf{NP}$-hardness extends to the case that $\ell=\infty$, i.e., for the problem of computing the minimum-enclosing ball under the Fréchet distance. Finally, we observe that a careful adaptation of Gonzalez' algorithm in combination with a curve simplification yields a $3$-approximation in any dimension, provided that an optimal simplification can be computed exactly. We conclude that our approximation bounds are close to being tight.

cs.CG↗