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Phillip Keldenich

Publications and source records attributed to Phillip Keldenich.

At least 19 recordsLinked to original sources
arXiv2026

Central Triangulation under Parallel Flip Operations: The CG:SHOP Challenge 2026

We give an overview of the 2026 Computational Geometry Challenge targeting the problem of finding a Central Triangulation under Parallel Flip Operations in triangulations of point sets. A flip is the parallel exchange of a set of edges in a triangulation with opposing diagonals of the convex quadrilaterals containing them. The challenge objective was, given a set of triangulations of a fixed point set, to determine a central triangulation with respect to parallel flip distances. More precisely, this asks for a triangulation that minimizes the sum of flip distances to all elements of the input

cs.CG↗
arXiv2025

Efficient Heuristics and Exact Methods for Pairwise Interaction Sampling

We consider a class of optimization problems that are fundamental to testing in modern configurable software systems, e.g., in automotive industries. In pairwise interaction sampling, we are given a (potentially very large) configuration space, in which each dimension corresponds to a possible Boolean feature of a software system; valid configurations are the satisfying assignments of a given propositional formula $φ$. The objective is to find a minimum-sized family of configurations, such that each pair of features is jointly tested at least once. Due to its relevance in Software Engineering, this problem has been studied extensively for over 20 years. In addition to new theoretical insights (we prove BH-hardness), we provide a broad spectrum of key contributions on the practical side that allow substantial progress for the practical performance. Remarkably, we are able to solve the largest instances we found in published benchmark sets (with about 500000000 feasible interactions) to provable optimality. Previous approaches were not even able to compute feasible solutions.

cs.DS↗
arXiv2025

Minimum Non-Obtuse Triangulations: The CG:SHOP Challenge 2025

We give an overview of the 2025 Computational Geometry Challenge targeting the problem Minimum Non-Obtuse Triangulation: Given a planar straight-line graph G in the plane, defined by a set of points in the plane (representing vertices) and a set of non-crossing line segments connecting them (representing edges); the objective is to find a feasible non-obtuse triangulation that uses a minimum number of Steiner points. If no triangulation without obtuse triangles is found, the secondary objective is to minimize the number of obtuse triangles in the triangulation.

cs.CG↗
arXiv2025

Exact Algorithms for Minimum Dilation Triangulation

We provide a spectrum of new theoretical insights and practical results for finding a Minimum Dilation Triangulation (MDT), a natural geometric optimization problem of considerable previous attention: Given a set $P$ of $n$ points in the plane, find a triangulation $T$, such that a shortest Euclidean path in $T$ between any pair of points increases by the smallest possible factor compared to their straight-line distance. No polynomial-time algorithm is known for the problem; moreover, evaluating the objective function involves computing the sum of (possibly many) square roots. On the other hand, the problem is not known to be NP-hard. (1) We provide practically robust methods and implementations for computing an MDT for benchmark sets with up to 30,000 points in reasonable time on commodity hardware, based on new geometric insights into the structure of optimal edge sets. Previous methods only achieved results for up to $200$ points, so we extend the range of optimally solvable instances by a factor of $150$. (2) We develop scalable techniques for accurately evaluating many shortest-path queries that arise as large-scale sums of square roots, allowing us to certify exact optimal solutions, with previous work relying on (possibly inaccurate) floating-point computations. (3) We resolve an open problem by establishing a lower bound of $1.44116$ on the dilation of the regular $84$-gon (and thus for arbitrary point sets), improving the previous worst-case lower bound of $1.4308$ and greatly reducing the remaining gap to the upper bound of $1.4482$ from the literature. In the process, we provide optimal solutions for regular $n$-gons up to $n = 100$.

cs.CG↗
arXiv2025

How Low Can We Go? Minimizing Interaction Samples for Configurable Systems

Modern software systems are typically configurable, a fundamental prerequisite for wide applicability and reusability. This flexibility poses an extraordinary challenge for quality assurance, as the enormous number of possible configurations makes it impractical to test each of them separately. This is where t-wise interaction sampling can be used to systematically cover the configuration space and detect unknown feature interactions. Over the last two decades, numerous algorithms for computing small interaction samples have been studied, providing improvements for a range of heuristic results; nevertheless, it has remained unclear how much these results can still be improved. We present a significant breakthrough: a fundamental framework, based on the mathematical principle of duality, for combining near-optimal solutions with provable lower bounds on the required sample size. This implies that we no longer need to work on heuristics with marginal or no improvement, but can certify the solution quality by establishing a limit on the remaining gap; in many cases, we can even prove optimality of achieved solutions. This theoretical contribution also provides extensive practical improvements: Our algorithm SampLNS was tested on 47 small and medium-sized configurable systems from the existing literature. SampLNS can reliably find samples of smaller size than previous methods in 85% of the cases; moreover, we can achieve and prove optimality of solutions for 63% of all instances. This makes it possible to avoid cumbersome efforts of minimizing samples by researchers as well as practitioners, and substantially save testing resources for most configurable systems.

cs.SE↗
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

Maximum Polygon Packing: The CG:SHOP Challenge 2024

We give an overview of the 2024 Computational Geometry Challenge targeting the problem \textsc{Maximum Polygon Packing}: Given a convex region $P$ in the plane, and a collection of simple polygons $Q_1, \ldots, Q_n$, each $Q_i$ with a respective value $c_i$, find a subset $S \subseteq \{1, \ldots,n\}$ and a feasible packing within $P$ of the polygons $Q_i$ (without rotation) for $i \in S$, maximizing $\sum_{i \in S} c_i$. Geometric packing problems, such as this, present significant computational challenges and are of substantial practical importance.

cs.CG↗
arXiv2023

Connected Coordinated Motion Planning with Bounded Stretch

We consider the problem of connected coordinated motion planning for a large collective of simple, identical robots: From a given start grid configuration of robots, we need to reach a desired target configuration via a sequence of parallel, collision-free robot motions, such that the set of robots induces a connected grid graph at all integer times. The objective is to minimize the makespan of the motion schedule, i.e., to reach the new configuration in a minimum amount of time. We show that this problem is NP-complete, even for deciding whether a makespan of 2 can be achieved, while it is possible to check in polynomial time whether a makespan of 1 can be achieved. On the algorithmic side, we establish simultaneous constant-factor approximation for two fundamental parameters, by achieving constant stretch for constant scale. Scaled shapes (which arise by increasing all dimensions of a given object by the same multiplicative factor) have been considered in previous seminal work on self-assembly, often with unbounded or logarithmic scale factors; we provide methods for a generalized scale factor, bounded by a constant. Moreover, our algorithm achieves a constant stretch factor: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of $d$, then the total duration of our overall schedule is $\mathcal{O}(d)$, which is optimal up to constant factors.

cs.CG↗
arXiv2023

Minimum Coverage by Convex Polygons: The CG:SHOP Challenge 2023

We give an overview of the 2023 Computational Geometry Challenge targeting the problem Minimum Coverage by Convex Polygons, which consists of covering a given polygonal region (possibly with holes) by a minimum number of convex subsets, a problem with a long-standing tradition in Computational Geometry.

cs.CG↗
arXiv2022

Packing Squares into a Disk with Optimal Worst-Case Density

We provide a tight result for a fundamental problem arising from packing squares into a circular container: The critical density of packing squares into a disk is $δ=\frac{8}{5π}\approx 0.509$. This implies that any set of (not necessarily equal) squares of total area $A \leq \frac{8}{5}$ can always be packed into a disk with radius 1; in contrast, for any $\varepsilon>0$ there are sets of squares of total area $\frac{8}{5}+\varepsilon$ that cannot be packed, even if squares may be rotated. This settles the last (and arguably, most elusive) case of packing circular or square objects into a circular or square container: The critical densities for squares in a square $\left(\frac{1}{2}\right)$, circles in a square $\left(\fracπ{(3+2\sqrt{2})}\approx 0.539\right)$ and circles in a circle $\left(\frac{1}{2}\right)$ have already been established, making use of recursive subdivisions of a square container into pieces bounded by straight lines, or the ability to use recursive arguments based on similarity of objects and container; neither of these approaches can be applied when packing squares into a circular container. Our proof uses a careful manual analysis, complemented by a computer-assisted part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms. At the same time, our approach showcases the power of a general framework for computer-assisted proofs, based on interval arithmetic.

cs.CG↗
arXiv2021

Area-Optimal Simple Polygonalizations: The CG Challenge 2019

We give an overview of theoretical and practical aspects of finding a simple polygon of minimum (Min-Area) or maximum (Max-Area) possible area for a given set of n points in the plane. Both problems are known to be NP-hard and were the subject of the 2019 Computational Geometry Challenge, which presented the quest of finding good solutions to more than 200 instances, ranging from n = 10 all the way to n = 1, 000, 000.

cs.CG↗
arXiv2021

Computing Area-Optimal Simple Polygonizations

We consider methods for finding a simple polygon of minimum (Min-Area) or maximum (Max-Area) possible area for a given set of points in the plane. Both problems are known to be NP-hard; at the center of the recent CG Challenge, practical methods have received considerable attention. However, previous methods focused on heuristic methods, with no proof of optimality. We develop exact methods, based on a combination of geometry and integer programming. As a result, we are able to solve instances of up to n=25 points to provable optimality. While this extends the range of solvable instances by a considerable amount, it also illustrates the practical difficulty of both problem variants.

cs.CG↗
arXiv2020

Worst-Case Optimal Covering of Rectangles by Disks

We provide the solution for a fundamental problem of geometric optimization by giving a complete characterization of worst-case optimal disk coverings of rectangles: For any $λ\geq 1$, the critical covering area $A^*(λ)$ is the minimum value for which any set of disks with total area at least $A^*(λ)$ can cover a rectangle of dimensions $λ\times 1$. We show that there is a threshold value $λ_2 = \sqrt{\sqrt{7}/2 - 1/4} \approx 1.035797\ldots$, such that for $λ<λ_2$ the critical covering area $A^*(λ)$ is $A^*(λ)=3π\left(\frac{λ^2}{16} +\frac{5}{32} + \frac{9}{256λ^2}\right)$, and for $λ\geq λ_2$, the critical area is $A^*(λ)=π(λ^2+2)/4$; these values are tight. For the special case $λ=1$, i.e., for covering a unit square, the critical covering area is $\frac{195π}{256}\approx 2.39301\ldots$. The proof uses a careful combination of manual and automatic analysis, demonstrating the power of the employed interval arithmetic technique.

cs.CG↗
arXiv2019

Parallel Online Algorithms for the Bin Packing Problem

We study \emph{parallel} online algorithms: For some fixed integer $k$, a collective of $k$ parallel processes that perform online decisions on the same sequence of events forms a $k$-\emph{copy algorithm}. For any given time and input sequence, the overall performance is determined by the best of the $k$ individual total results. Problems of this type have been considered for online makespan minimization; they are also related to optimization with \emph{advice} on future events, i.e., a number of bits available in advance. We develop \textsc{Predictive Harmonic}$_3$ (PH3), a relatively simple family of $k$-copy algorithms for the online Bin Packing Problem, whose joint competitive factor converges to 1.5 for increasing $k$. In particular, we show that $k=6$ suffices to guarantee a factor of $1.5714$ for PH3, which is better than $1.57829$, the performance of the best known 1-copy algorithm \textsc{Advanced Harmonic}, while $k=11$ suffices to achieve a factor of $1.5406$, beating the known lower bound of $1.54278$ for a single online algorithm. In the context of online optimization with advice, our approach implies that 4 bits suffice to achieve a factor better than this bound of $1.54278$, which is considerably less than the previous bound of 15 bits.

cs.DS↗
arXiv2019

Packing Disks into Disks with Optimal Worst-Case Density

We provide a tight result for a fundamental problem arising from packing disks into a circular container: The critical density of packing disks in a disk is 0.5. This implies that any set of (not necessarily equal) disks of total area $δ\leq 1/2$ can always be packed into a disk of area 1; on the other hand, for any $\varepsilon>0$ there are sets of disks of area $1/2+\varepsilon$ that cannot be packed. The proof uses a careful manual analysis, complemented by a minor automatic part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms.

cs.CG↗