Travelling Salesman Problem with Neighborhoods

Práce se zabývá využitím metaheuristického algoritmu GLNS, používaného k řešení problému obecného obchodního cestujícího, k řešení upraveného problému obchodního cestujícího se sousedstvími. Tato úprava spočívá v tom, že sousedstvími jsou pouze nedegenerované mnohoúhelníky, jež se mohou i překrývat. V rámci práce jsou navrhnuty a implementovány dva algorithmy, které využívají původní nebo modifikovaný algoritmus GLNS. Dále je v obou také využit algoritmus pro řešení úlohy průchodu mnohoúhelníky. Druhý navrhnutý algoritmus je schopný řešit i instance, kde jsou mezi sousedstvími překážky ve tvaru nedegenerovaných mnohoúhelníků. Využívá k tomu datové struktury, která se nazývá 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

The Visibility Freeze-Tag Problem

In the Freeze-Tag Problem, we are given a set of robots at points inside some metric space. Initially, all the robots are frozen except one. That robot can awaken (or “unfreeze”) another robot by moving to its position, and once a robot is awakened, it can move and help to awaken other robots. The goal is to awaken all the robots in the shortest time. The Freeze-Tag Problem has been studied in different metric spaces: graphs and Euclidean spaces. In this thesis, we look at the Freeze-Tag Problem in polygons, and we introduce the Visibility Freeze-Tag Problem, where one robot can awaken another robot by “seeing” it. Furthermore, we introduce a variant of the Visibility Freeze-Tag Problem, called the Line/Point Freeze Tag Problem, where each robot lies on an awakening line, and one robot can awaken another robot by touching its awakening line. We survey the current results for the Freeze-Tag Problem in graphs, Euclidean spaces and polygons. Since the Visibility Freeze-Tag Problem bears some resemblance to the Watchman Route Problem, we also survey the background literature on the Watchman Route Problem. We show that the Freeze-Tag Problem in polygons and the Visibility Freeze-Tag Problem are NP-hard, and we present an O(n)-approximation algorithm for the Visibility Freeze-Tag Problem. For the Line/Point Freeze-Tag Problem, we give a polynomial time algorithm for the special case where all the awakening lines are parallel to each other. We prove that the general case is NP-hard, and we present an O(1)- approximation algorithm

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