arXiv subjects

Rostislav Staněk

Publications and source records attributed to Rostislav Staněk.

3 recordsLinked to original sources

arXiv2026

A new O(n log n) approach for the Euclidean maximum weight matching problem

In a weighted graph $G = (V, E)$, the maximum weight matching problem (MWM) asks for a matching (i.e. pairing) of its vertices, such that each vertex is paired with at most one other vertex and the total sum of weights of all edges connecting paired vertices is maximised. If the vertices of the graph correspond to points in the Euclidean plane and the weights to their pairwise Euclidean distances, we get the Euclidean maximum weight matching problem (Euclidean MWM). The best optimum-solution algorithm for this problem runs in $O(n^{2.5})$. Furthermore, there exists an FPTAS guaranteeing a $(1 - ε)$-approximation ratio, which runs in $O(m ε^{-1} \log ε^{-1})$ time. Heuristics with a subquadratic running time (with respect to the number of vertices $|V|$) are known, but often yield solutions of a modest quality. In this paper, we present a novel algorithm for solving the Euclidean MWM running in $O(n \log n)$ time and providing excellent solution quality, especially for larger instances. In particular, in our computational tests, the algorithm yielded optimum or near-optimum solutions for all test instances; the worst observed optimality gap was less than $1.38\%$. This makes the algorithm highly attractive for practical applications, especially when exact methods become computationally prohibitive due to the size of the instance.

cs.CG↗

arXiv2023

Advanced optimization models for the location of charging stations in e-mobility

For a reduction in environmental pollution and dependency on petroleum, electric vehicles (EV) present an advantageous alternative to traditionally fossil-fuel powered automobiles. Rapid growth in the number of EVs requires an urgent need to develop an adequate charging station infrastructure to stimulate and facilitate their usage. Due to restricted investments in the development of a sufficient infrastructure, locations have to be chosen deliberately. In this paper, three extensions considering different objectives and various constraints to the deterministic flow refuelling location problem (DFRLP), described 2017 by de Vries and Duijzer, are introduced. In the first extension we ask how many charging stations (CS) are necessary to cover a pre-specified number of EVs and therefore exchange the original objective function for a minimizing cost function. Secondly, our research shows that, when considering location-dependent construction costs, results heavily depend on the relations of said cost differences. Tests for different cost scenarios are carried out and policy implications are discussed. In the last extension, we consider the capacity of a CS to be limited. The DFRLP assumes an unlimited capacity, meaning it is always possible to refuel all EVs at all CSs, where they stop. In our model the capacity is put into relation to the total sum of demands generated by all EVs, passing a particular CS, which means that our model determines the placement and the sizes of all CSs simultaneously. Finally, all extensions are evaluated using benchmarks instances based on test instances from the literature.

math.OC↗

arXiv2018

Geometric and LP-based heuristics for the quadratic travelling salesman problem

A generalization of the classical TSP is the so-called quadratic travelling salesman problem (QTSP), in which a cost coefficient is associated with the transition in every vertex, i.e. with every pair of edges traversed in succession. In this paper we consider two geometrically motivated special cases of the QTSP known from the literature, namely the angular-metric TSP, where transition costs correspond to turning angles in every vertex, and the angular-distance-metric TSP, where a linear combination of turning angles and Euclidean distances is considered. At first we introduce a wide range of heuristic approaches, motivated by the typical geometric structure of optimal solutions. In particular, we exploit lens-shaped neighborhoods of edges and a decomposition of the graph into layers of convex hulls, which are then merged into a tour by a greedy-type procedure or by utilizing an ILP model. Secondly, we consider an ILP model for a standard linearization of QTSP and compute fractional solutions of a relaxation. By rounding we obtain a collection of subtours, paths and isolated points, which are combined into a tour by various strategies, all of them involving auxiliary ILP models. Finally, different improvement heuristics are proposed, most notably a matheuristic which locally reoptimizes the solution for rectangular sectors of the given point set by an ILP approach. Extensive computational experiments for benchmark instances from the literature and extensions thereof illustrate the Pareto-efficient frontier of algorithms in a (running time, objective value)-space. It turns out that our new methods clearly dominate the previously published heuristics.

cs.DM↗