Johnson, Room 2D-150, AT&T Bell Laboratories, Murray Hill, NJ 07974 (or to Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should send them to David S. A background equivalent to that provided by is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Freeman & Co., New York, 1979 (hereinafter referred to as "" previous columns will be referred to by their dates). Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness," W. The presentation is modeled on that used by M. This is the 23rd edition of an irregularly appearing column that covers new developments in the theory of NP-completeness. Our main new results are that the metric dimension of a graph with n nodes can be approximated in polynomial time within a factor of O(logn), and some properties of graphs with metric dimension two. In this paper we present some results about this problem. A minimum set of landmarks which uniquely determine the robot's position is called a “metric basis”, and the minimum number of landmarks is called the “metric dimension” of the graph. This suggests the following problem: given a graph, what are the fewest number of landmarks needed, and where should they be located, so that the distances to the landmarks uniquely determine the robot's position on the graph? This is actually a classical problem about metric spaces. Instead, we shall assume that a robot navigating on a graph can sense the distances to a set of landmarks.Evidently, if the robot knows its distances to a sufficiently large set of landmarks, its position on the graph is uniquely determined. On a graph, however, there is neither the concept of direction nor that of visibility. For a robot navigating in Euclidean space, visual detection of a distinctive landmark provides information about the direction to the landmark, and allows the robot to determine its position by triangulation. The robot can locate itself by the presence of distinctively labeled “landmark” nodes in the graph space. Navigation can be studied in a graph-structured framework in which the navigating agent (which we shall assume to be a point robot) moves from node to node of a “graph space”. We demonstrate these properties using extensive simulation studies. Finally, we propose a routingĪlgorithm that uses geographic routing in the greedy phase and virtual coordinates with backtracking to overcome voids andĪchieve high connectivity in the greedy phase with higher overall path quality and more resilience to localization errors. It also compares several existing routing protocols based on Virtual Coordinate systems. In addition, it is possible for nodes with the sameĬoordinates to arise at different points in the network in the presence of voids. Where packets reach nodes with no viable next hop in the forwarding set. However, we show that it is vulnerable to different forms of the void problem That is more resilient to localization errors. Virtual coordinate systems which overlay a coordinate system on the nodes offer an alternative Geographical routing protocols have several desirable features for use in ad hoc and sensor networks but are susceptible to In this case the presented approach reaches optimal or best known solutions for hypercubes up to 131072 nodes and We have also modified our implementation to handle theoretically challenging large-scale classes of hypercubes and Hamming For smaller instances, GA solutions are compared with CPLEX results. GA relatively quickly obtains approximate solutions. Testing on instances with up to 1534 nodes shows that NP-hard problems: pseudo boolean, crew scheduling and graph coloring. For that reason, we present the results of the computational experience on several sets of test instances for other Metric dimension problem up to now has been considered only theoretically, standard test instances for this problem do notĮxist. The overall performance of the GA implementation is improved by a caching technique. The feasibility is enforcedīy repairing the individuals. (GA) that uses the binary encoding and the standard genetic operators adapted to the problem. (It could do with a clean-up with regard to temporary variables, and a few styling options would be a reasonable addition.In this paper we consider the NP-hard problem of determining the metric dimension of graphs. I use LaTeX3 to do some conversions between integers and binary, and to make the various recursions easier. I'm going to interpret "hypercube" as a way of specifying a graph.
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