arXiv subjects

Sándor Fekete

Publications and source records attributed to Sándor Fekete.

5 recordsLinked to original sources

arXiv2026

Drone Air Traffic Control: Tracking a Set of Moving Objects with Minimal Power

A common sensing problem is to use a set of stationary tracking locations to monitor a collection of moving devices: Given $n$ objects that need to be tracked, each following its own trajectory, and $m$ stationary traffic control stations, each with a sensing region of adjustable range; how should we adjust the individual sensor ranges in order to optimize energy consumption? We provide both negative theoretical and positive practical results for this important and natural challenge. On the theoretical side, we show that even if all objects move at constant speed along straight lines, no polynomial-time algorithm can guarantee optimal coverage for a given starting solution. On the practical side, we present an algorithm based on geometric insights that is able to find optimal solutions for the $\min \max$ variant of the problem, which aims at minimizing peak power consumption. Runtimes for instances with 500 moving objects and 25 stations are in the order of seconds for scenarios that take minutes to play out in the real world, demonstrating real-time capability of our methods.

cs.CG↗

arXiv2025

A quantum search method for quadratic and multidimensional knapsack problems

Solving combinatorial optimization problems is a promising application area for quantum algorithms in real-world scenarios. In this work, we extend the "Quantum Tree Generator" (QTG), previously proposed for the 0-1 Knapsack Problem, to the 0-1 Quadratic Knapsack Problem (QKP) and the Multidimensional Knapsack Problem (MDKP). The QTG constructs a superposition of all feasible solutions for a given instance and can therefore be utilized as a promising state preparation routine within amplitude amplification to produce high-quality solutions. Previously, QTG-based search was tested on the 0-1 Knapsack Problem, where it demonstrated the potential for practical quantum advantage, once quantum computers with a few hundred logical and fully connected qubits are available. Here, we evaluate the algorithm's performance on QKP and MDKP against the classical solver Gurobi. To facilitate large-scale evaluations, we employ an advanced benchmarking technique that enables runtime predictions for instances with up to 2000 variables for QKP and up to 1500 variables and 100 constraints for MDKP. Our results indicate that QTG-based search can produce high-quality solutions with competitive runtimes for QKP. However, its performance declines for MDKP, highlighting the challenges quantum algorithms face when tackling highly constrained optimization problems.

quant-ph↗

arXiv2024

A quantum algorithm for solving 0-1 Knapsack problems

Here we present two novel contributions for achieving quantum advantage in solving difficult optimisation problems, both in theory and foreseeable practice. (1) We introduce the "Quantum Tree Generator", an approach to generate in superposition all feasible solutions of a given instance, yielding together with amplitude amplification the optimal solutions for 0-1 knapsack problems. The QTG offers massive memory savings and enables competitive runtimes compared to the classical state-of-the-art knapsack solvers (such as COMBO, Gurobi, CP-SAT, Greedy) already for instances involving as few as 100 variables. (2) By introducing a new runtime calculation technique that exploits logging data from the classical solver COMBO, we can predict the runtime of our method way beyond the range of existing quantum platforms and simulators, for various benchmark instances with up to 600 variables. Combining both of these innovations, we demonstrate the QTG's potential practical quantum advantage for large-scale problems, indicating an effective approach for combinatorial optimisation problems.

quant-ph↗

arXiv2019

Existence and hardness of conveyor belts

An open problem of Manuel Abellanas asks whether every set of disjoint closed unit disks in the plane can be connected by a conveyor belt, which means a tight simple closed curve that touches the boundary of each disk, possibly multiple times. We prove three main results. First, for unit disks whose centers are both $x$-monotone and $y$-monotone, or whose centers have $x$-coordinates that differ by at least two units, a conveyor belt always exists and can be found efficiently. Second, it is NP-complete to determine whether disks of varying radii have a conveyor belt, and it remains NP-complete when we constrain the belt to touch disks exactly once. Third, any disjoint set of $n$ disks of arbitrary radii can be augmented by $O(n)$ "guide" disks so that the augmented system has a conveyor belt touching each disk exactly once, answering a conjecture of Demaine, Demaine, and Palop.

cs.CG↗

arXiv2016

Connectivity graphs of uncertainty regions

We study connectivity relations among points, where the precise location of each input point lies in a region of uncertainty. We distinguish two fundamental scenarios under which uncertainty arises. In the favorable Best-Case Uncertainty (BU), each input point can be chosen from a given set to yield the best possible objective value. In the unfavorable Worst-Case Uncertainty (WU), the input set has worst possible objective value among all possible point locations, which are uncertain due, for example, to imprecise data. We consider these notions of uncertainty for the bottleneck spanning tree problem, giving rise to the following Best-Case Connectivity with Uncertainty (BCU) problem: Given a family of geometric regions, choose one point per region, such that the longest edge length of an associated geometric spanning tree is minimized. We show that this problem is NP-hard even for very simple scenarios in which the regions are line segments or squares. On the other hand, we give an exact solution for the case in which there are n+k regions, where k of the regions are line segments and n of the regions are fixed points. We then give approximation algorithms for cases where the regions are either all line segments or all unit discs. We also provide approximation methods for the corresponding Worst-Case Connectivity with Uncertainty (WCU) problem: Given a set of uncertainty regions, find the minimal distance r such that for any choice of points, one per region, there is a spanning tree among the points with edge length at most r.

cs.CG↗