arXiv subjects

Sándor P. Fekete

Publications and source records attributed to Sándor P. Fekete.

At least 19 recordsLinked to original sources

arXiv2026

Pass the Bucket: Efficient, Robust, Local Load Balancing for Teams of Heterogeneous Robots

We study the problem of decentralized, self-organized task sharing for a swarm of heterogeneous robots that collaborate in transportation or other objectives that require coordinated motion planning. To this end, we present theoretical and practical results for the simple but effective mechanism of \emph{bucket brigades} for load balancing, in which a team of heterogenous robots share a spatial task in a confined, one-dimensional space, while only being able to sense collisions with neighbors or walls. The goal is to optimize throughput of the overall system, without central control or information, aiming at an interval partition proportional to robot velocities. We address possible chaotic system behavior by developing a stabilization mechanism based on simple local aid, a ``token'', that temporarily decelerates robots after an encounter. This purely local change eliminates persistent oscillations, resulting in convergence towards a stable system state. We accelerate system convergence by comparing a single boundary token to ubiquitous two-directional tokens and optimizing the deceleration factor. Event-driven simulations report convergence times and robustness: For a large variety of perturbations (such as robot deletion, position or velocity jittering), the system reliably re-converges. The results suggest a local, practical mechanism for robust load balancing for heterogeneous teams of robots that promises an effective tool as basis for more complex scenarios.

cs.RO↗

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↗

arXiv2026

Tilt Automata: Gathering Particles With Uniform External Control

Motivated by targeted drug delivery, we investigate the gathering of particles in the full tilt model of externally controlled motion planning: A set of particles is located at the tiles of a polyomino with all particles reacting uniformly to an external force by moving as far as possible in one of the four axis-parallel directions until they hit the boundary. The goal is to choose a sequence of directions that moves all particles to a common position. Our results include a polynomial-time algorithm for gathering in a completely filled polyomino as well as hardness reductions for approximating shortest gathering sequences and for determining whether the particles in a partially filled polyomino can be gathered. We pay special attention to the impact of restricted geometry, particularly polyominoes without holes. As corollaries, we make progress on an open question from [Balanza-Martinez et al., SODA 2020] by showing that deciding whether a given position can be occupied remains NP-hard in polyominoes without holes and provide initial results on the parameterized complexity of tilt problems. Our results build on a connection we establish between tilt models and the theory of synchronizing automata.

cs.CG↗

arXiv2025

Sliding Squares in Parallel

We consider algorithmic problems motivated by modular robotic reconfiguration in the sliding square model, in which we are given $n$ square-shaped modules in a (labeled or unlabeled) start configuration and need to find a schedule of sliding moves to transform it into a desired goal configuration, maintaining connectivity of the configuration at all times. Recent work has aimed at minimizing the total number of moves, resulting in fully sequential schedules that perform reconfiguration in $\mathcal{O}(nP)$ moves for arrangements of bounding box perimeter size $P$, or a number of moves linear in the sum of module coordinates in the start and target arrangements. We extend the model to leverage the possibility of parallel motion, thereby reducing worst-case makespans by a factor linear in $n$. Our work presents tight results both in terms of complexity and algorithms: We show that deciding the existence of a single parallel reconfiguration step that solves an instance is NP-complete for unlabeled modules, but can be solved efficiently in the labeled setting. Nevertheless, deciding whether a labeled instance can be solved in two parallel steps is NP-complete. Finally, we describe an algorithm to perform in-place reconfiguration in worst-case optimal $\mathcal{O}(P)$ parallel steps for the unlabeled setting. This algorithm has a straight-forward extension to the labeled setting with slight relaxations to either the reconfiguration time or space constraint.

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

Moving Matter: Using a Single, Simple Robot to Reconfigure a Connected Set of Building Blocks

We implement and evaluate different methods for the reconfiguration of a connected arrangement of tiles into a desired target shape, using a single active robot that can move along the tile structure. This robot can pick up, carry, or drop off one tile at a time, but it must maintain a single connected configuration at all times. Becker et al. (CCCG 2025) recently proposed an algorithm that uses histograms as canonical intermediate configurations, guaranteeing performance within a constant factor of the optimal solution if the start and target configuration are well-separated. We implement and evaluate this algorithm, both in a simulated and practical setting, using an inchworm type robot to compare it with two existing heuristic algorithms.

cs.RO↗

arXiv2025

Efficient Reconfiguration of Tile Arrangements by a Single Active Robot

We consider the problem of reconfiguring a two-dimensional connected grid arrangement of passive building blocks from a start configuration to a goal configuration, using a single active robot that can move on the tiles, remove individual tiles from a given location and physically move them to a new position by walking on the remaining configuration. The objective is to determine a schedule that minimizes the overall makespan, while keeping the tile configuration connected. We provide both negative and positive results. (1) We generalize the problem by introducing weighted movement costs, which can vary depending on whether tiles are carried or not, and prove that this variant is NP-hard. (2) We give a polynomial-time constant-factor approximation algorithm for the case of disjoint start and target bounding boxes, which additionally yields optimal carry distance for 2-scaled instances.

cs.CG↗

arXiv2025

Guarding Offices with Maximum Dispersion

We investigate the Dispersive Art Gallery Problem with vertex guards and rectangular visibility ($r$-visibility) for a class of orthogonal polygons that reflect the properties of real-world floor plans: these office-like polygons consist of rectangular rooms and corridors. In the dispersive variant of the Art Gallery Problem, the objective is not to minimize the number of guards but to maximize the minimum geodesic $L_1$-distance between any two guards, called the dispersion distance. Our main contributions are as follows. We prove that determining whether a vertex guard set can achieve a dispersion distance of $4$ in office-like polygons is NP-complete, where vertices of the polygon are restricted to integer coordinates. Additionally, we present a simple worst-case optimal algorithm that guarantees a dispersion distance of $3$ in polynomial time. Our complexity result extends to polyominoes, resolving an open question posed by Rieck and Scheffer (CGTA 2024). When vertex coordinates are allowed to be rational, we establish analogous results, proving that achieving a dispersion distance of $2+\varepsilon$ is NP-hard for any $\varepsilon > 0$, while the classic Art Gallery Problem remains solvable in polynomial time for this class of polygons. Furthermore, we give a straightforward polynomial-time algorithm that computes worst-case optimal solutions with a dispersion distance of $2$. On the other hand, for the more restricted class of hole-free independent office-like polygons, we propose a dynamic programming approach that computes optimal solutions. Moreover, we demonstrate that the problem is practically tractable for arbitrary orthogonal polygons. To this end, we compare solvers based on SAT, CP, and MIP formulations. Notably, SAT solvers efficiently compute optimal solutions for randomly generated instances with up to $1600$ vertices in under $15$s.

cs.CG↗

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

Drainability and Fillability of Polyominoes in Diverse Models of Global Control

Tilt models offer intuitive and clean definitions of complex systems in which particles are influenced by global control commands. Despite a wide range of applications, there has been almost no theoretical investigation into the associated issues of filling and draining geometric environments. This is partly because a globally controlled system (i.e., passive matter) exhibits highly complex behavior that cannot be locally restricted. Thus, there is a strong need for theoretical studies that investigate these models both (1) in terms of relative power to each other, and (2) from a complexity theory perspective. In this work, we provide (1) general tools for comparing and contrasting different models of global control, and (2) both complexity and algorithmic results on filling and draining.

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

Multi-Covering a Point Set by $m$ Disks with Minimum Total Area

A common robotics sensing problem is to place sensors to robustly monitor a set of assets, where robustness is assured by requiring asset $p$ to be monitored by at least $κ(p)$ sensors. Given $n$ assets that must be observed by $m$ sensors, each with a disk-shaped sensing region, where should the sensors be placed to minimize the total area observed? We provide and analyze a fast heuristic for this problem. We then use the heuristic to initialize an exact Integer Programming solution. Subsequently, we enforce separation constraints between the sensors by modifying the integer program formulation and by changing the disk candidate set.

cs.RO↗

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

Coordinated Motion Planning: Multi-Agent Path Finding in a Densely Packed, Bounded Domain

We study Multi-Agent Path Finding for arrangements of labeled agents in the interior of a simply connected domain: Given a unique start and target position for each agent, the goal is to find a sequence of parallel, collision-free agent motions that minimizes the overall time (the makespan) until all agents have reached their respective targets. A natural case is that of a simply connected polygonal domain with axis-parallel boundaries and integer coordinates, i.e., a simple polyomino, which amounts to a simply connected union of lattice unit squares or cells. We focus on the particularly challenging setting of densely packed agents, i.e., one per cell, which strongly restricts the mobility of agents, and requires intricate coordination of motion. We provide a variety of novel results for this problem, including (1) a characterization of polyominoes in which a reconfiguration plan is guaranteed to exist; (2) a characterization of shape parameters that induce worst-case bounds on the makespan; (3) a suite of algorithms to achieve asymptotically worst-case optimal performance with respect to the achievable stretch for cases with severely limited maneuverability. This corresponds to bounding the ratio between obtained makespan and the lower bound provided by the max-min distance between the start and target position of any agent and our shape parameters. Our results extend findings by Demaine et al. (SIAM Journal on Computing, 2019) who investigated the problem for solid rectangular domains, and in the closely related field of Permutation Routing, as presented by Alpert et al. (Computational Geometry, 2022) for convex pieces of grid graphs.

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

Efficiently Reconfiguring a Connected Swarm of Labeled Robots

When considering motion planning for a swarm of $n$ labeled robots, we need to rearrange a given start configuration into a desired target configuration via a sequence of parallel, collision-free robot motions. The objective is to reach the new configuration in a minimum amount of time; an important constraint is to keep the swarm connected at all times. Problems of this type have been considered before, with recent notable results achieving constant stretch for not necessarily connected reconfiguration: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of $d$, the total duration of an overall schedule can be bounded to $\mathcal{O}(d)$, which is optimal up to constant factors. However, constant stretch could only be achieved if disconnected reconfiguration is allowed, or for scaled configurations (which arise by increasing all dimensions of a given object by the same multiplicative factor) of unlabeled robots. We resolve these major open problems by (1) establishing a lower bound of $Ω(\sqrt{n})$ for connected, labeled reconfiguration and, most importantly, by (2) proving that for scaled arrangements, constant stretch for connected reconfiguration can be achieved. In addition, we show that (3) it is NP-complete to decide 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.

cs.CG↗

arXiv2024

Dispersive Vertex Guarding for Simple and Non-Simple Polygons

We study the Dispersive Art Gallery Problem with vertex guards: Given a polygon $\mathcal{P}$, with pairwise geodesic Euclidean vertex distance of at least $1$, and a rational number $\ell$; decide whether there is a set of vertex guards such that $\mathcal{P}$ is guarded, and the minimum geodesic Euclidean distance between any two guards (the so-called dispersion distance) is at least $\ell$. We show that it is NP-complete to decide whether a polygon with holes has a set of vertex guards with dispersion distance $2$. On the other hand, we provide an algorithm that places vertex guards in simple polygons at dispersion distance at least $2$. This result is tight, as there are simple polygons in which any vertex guard set has a dispersion distance of at most $2$.

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↗