144 research outputs found
Travelling Salesman Problem with Neighborhoods
Práce se zabĂ˝vá vyuĹľitĂm metaheuristickĂ©ho algoritmu GLNS, pouĹľĂvanĂ©ho k Ĺ™ešenĂ problĂ©mu obecnĂ©ho obchodnĂho cestujĂcĂho, k Ĺ™ešenĂ upravenĂ©ho problĂ©mu obchodnĂho cestujĂcĂho se sousedstvĂmi. Tato Ăşprava spoÄŤĂvá v tom, Ĺľe sousedstvĂmi jsou pouze nedegenerovanĂ© mnohoĂşhelnĂky, jeĹľ se mohou i pĹ™ekrĂ˝vat. V rámci práce jsou navrhnuty a implementovány dva algorithmy, kterĂ© vyuĹľĂvajĂ pĹŻvodnĂ nebo modifikovanĂ˝ algoritmus GLNS. Dále je v obou takĂ© vyuĹľit algoritmus pro Ĺ™ešenĂ Ăşlohy prĹŻchodu mnohoĂşhelnĂky. DruhĂ˝ navrhnutĂ˝ algoritmus je schopnĂ˝ Ĺ™ešit i instance, kde jsou mezi sousedstvĂmi pĹ™ekážky ve tvaru nedegenerovanĂ˝ch mnohoĂşhelnĂkĹŻ. VyuĹľĂvá k tomu datovĂ© struktury, která se nazĂ˝vá graf viditelnosti.This thesis explores the possibility of transforming the metaheuristic algorithm GLNS, used for General Travelling Salesman Problem (GTSP), to instead solve the version of Travelling Salesman Problem with Neighborhoods (TSPN) where the neighborhoods are simple, possibly intersecting, polygons. Two algorithms are proposed and implemented, each utilizing GLNS in a different way. Both also make use of an algorithm solving the unconstrained version of Touring Polygons Problem (TPP). The second proposed algorithm is additionally equipped to handle a case, when there are simple polygonal obstacles between the neighborhoods. This is made possible using a visibility graph
Computing Smallest Convex Intersecting Polygons
Funding Information: Funding Mark de Berg is supported by the Dutch Research Council (NWO) through Gravitation-grant NETWORKS-024.002.003. Antonis Skarlatos: Part of the work was done during an internship at the Max Planck Institute for Informatics in Saarbrücken, Germany. Publisher Copyright: © 2022 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.A polygon C is an intersecting polygon for a set O of objects in R2 if C intersects each object in O, where the polygon includes its interior. We study the problem of computing the minimum-perimeter intersecting polygon and the minimum-area convex intersecting polygon for a given set O of objects. We present an FPTAS for both problems for the case where O is a set of possibly intersecting convex polygons in the plane of total complexity n. Furthermore, we present an exact polynomial-time algorithm for the minimum-perimeter intersecting polygon for the case where O is a set of n possibly intersecting segments in the plane. So far, polynomial-time exact algorithms were only known for the minimum perimeter intersecting polygon of lines or of disjoint segments.Peer reviewe
Minimum cost b-matching problems with neighborhoods
In this paper, we deal with minimum cost b-matching problems on graphs where the nodes are assumed to belong to non-necessarily convex regions called neighborhoods, and the costs are given by the distances between points of the neighborhoods. The goal in the proposed problems is twofold: (i) finding a b-matching in the graph and (ii) determining a point in each neighborhood to be the connection point among the edges defining the b-matching. Different variants of the minimum cost b-matching problem are considered depending on the criteria to match neighborhoods: perfect, maximum cardinality, maximal and the a-b-matching problems. The theoretical complexity of solving each one of these problems is analyzed. Different mixed integer non-linear programming formulations are proposed for each one of the considered problems and then reformulated as Second Order Cone formulations. An extensive computational experience shows the efficiency of the proposed formulations to solve the problems under study
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
The Visibility Center of a Simple Polygon
We introduce the visibility center of a set of points inside a polygon - a point c_V such that the maximum geodesic distance from c_V to see any point in the set is minimized. For a simple polygon of n vertices and a set of m points inside it, we give an O((n+m) log (n+m)) time algorithm to find the visibility center. We find the visibility center of all points in a simple polygon in O(n log n) time.
Our algorithm reduces the visibility center problem to the problem of finding the geodesic center of a set of half-polygons inside a polygon, which is of independent interest. We give an O((n+k) log (n+k)) time algorithm for this problem, where k is the number of half-polygons
Approximation Algorithms for the Two-Watchman Route in a Simple Polygon
The two-watchman route problem is that of computing a pair of closed tours in
an environment so that the two tours together see the whole environment and
some length measure on the two tours is minimized. Two standard measures are:
the minmax measure, where we want the tours where the longest of them has
smallest length, and the minsum measure, where we want the tours for which the
sum of their lengths is the smallest. It is known that computing a minmax
two-watchman route is NP-hard for simple rectilinear polygons and thus also for
simple polygons. Also, any c-approximation algorithm for the minmax
two-watchman route is automatically a 2c-approximation algorithm for the minsum
two-watchman route. We exhibit two constant factor approximation algorithms for
computing minmax two-watchman routes in simple polygons with approximation
factors 5.969 and 11.939, having running times O(n^8) and O(n^4) respectively,
where n is the number of vertices of the polygon. We also use the same
techniques to obtain a 6.922-approximation for the fixed two-watchman route
problem running in O(n^2) time, i.e., when two starting points of the two tours
are given as input.Comment: 36 pages, 14 figure
Shortest Paths in Graphs of Convex Sets
Given a graph, the shortest-path problem requires finding a sequence of edges
with minimum cumulative length that connects a source to a target vertex. We
consider a generalization of this classical problem in which the position of
each vertex in the graph is a continuous decision variable, constrained to lie
in a corresponding convex set. The length of an edge is then defined as a
convex function of the positions of the vertices it connects. Problems of this
form arise naturally in road networks, robot navigation, and even optimal
control of hybrid dynamical systems. The price for such a wide applicability is
the complexity of this problem, which is easily seen to be NP-hard. Our main
contribution is a strong mixed-integer convex formulation based on perspective
functions. This formulation has a very tight convex relaxation and allows to
efficiently find globally-optimal paths in large graphs and in high-dimensional
spaces
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