74,680 research outputs found

    Computing Minimum Spanning Trees with Uncertainty

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    We consider the minimum spanning tree problem in a setting where information about the edge weights of the given graph is uncertain. Initially, for each edge ee of the graph only a set AeA_e, called an uncertainty area, that contains the actual edge weight wew_e is known. The algorithm can `update' ee to obtain the edge weight weAew_e \in A_e. The task is to output the edge set of a minimum spanning tree after a minimum number of updates. An algorithm is kk-update competitive if it makes at most kk times as many updates as the optimum. We present a 2-update competitive algorithm if all areas AeA_e are open or trivial, which is the best possible among deterministic algorithms. The condition on the areas AeA_e is to exclude degenerate inputs for which no constant update competitive algorithm can exist. Next, we consider a setting where the vertices of the graph correspond to points in Euclidean space and the weight of an edge is equal to the distance of its endpoints. The location of each point is initially given as an uncertainty area, and an update reveals the exact location of the point. We give a general relation between the edge uncertainty and the vertex uncertainty versions of a problem and use it to derive a 4-update competitive algorithm for the minimum spanning tree problem in the vertex uncertainty model. Again, we show that this is best possible among deterministic algorithms

    Narrow passage identification using cell decomposition approximation and minimum spanning tree

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    Narrow passage problem is a problematic issue facing the sampling-based motion planner. In this paper, a new approach for narrow areas identification is proposed. The quad-tree cell-decomposition approximation is used to divide the free workspace into smaller cells, and build a graph of adjacency for these. The proposed method follows the graph edges and finds a sequence of cells, which have the same size, preceded and followed by a bigger cell size. The sequence, which has the pattern bigger-smaller-bigger cells size, is more likely to be located in a narrow area. The minimum spanning tree algorithm is used, to linearize adjacency graph. Many methods have been proposed to manipulate the edges cost in the graph, in order to make the generated spanning tree traverse through narrow passages in detectable ways. Five methods have been proposed, some of them give bad results, and the others give better on in simulationsNarrow passage problem is a problematic issue facing the sampling-based motion planner. In this paper, a new approach for narrow areas identification is proposed. The quad-tree cell-decomposition approximation is used to divide the free workspace into smaller cells, and build a graph of adjacency for these. The proposed method follows the graph edges and finds a sequence of cells, which have the same size, preceded and followed by a bigger cell size. The sequence, which has the pattern bigger-smaller-bigger cells size, is more likely to be located in a narrow area. The minimum spanning tree algorithm is used, to linearize adjacency graph. Many methods have been proposed to manipulate the edges cost in the graph, in order to make the generated spanning tree traverse through narrow passages in detectable ways. Five methods have been proposed, some of them give bad results, and the others give better on in simulation

    The Minimum Labeling Spanning Tree Problem and Some Variants

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    The focus of my dissertation research involves combinatorial optimization. This is a key area in operations research and computer science. It includes lots of problems that have a wide variety of real-world applications. In addition, most of these problems are inherently difficult to solve. My specific disseration topic is the minimum labeling spanning tree (MLST) problem and some variants, including the label-constrained minimum spanning tree (LC-MST) problem and the colorful travaling salesman problem (CTSP). All of the three problems are NP-hard. The MLST problem tries to find a spanning tree of a graph with the smallest number of labels. The LC-MST problem tries to find the minimum-cost spanning tree of a graph with no more than K labels. The CTSP tries to find a hamiltonian cycle of a graph with the smallest number of labels. For each of the problems, we use both heuristic and genetic algorithms to solve them. From the computational results, the genetic algorithm can always obtain a better tradeoff between the solution quality and the running time. My disseration research shows that the genetic algorithm can be successfully applied to solve many NP-hard problems

    Equidistribution of Point Sets for the Traveling Salesman and Related Problems

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    Given a set S of n points in the unit square [0, 1)2, an optimal traveling salesman tour of S is a tour of S that is of minimum length. A worst-case point set for the Traveling Salesman Problem in the unit square is a point set S(n) whose optimal traveling salesman tour achieves the maximum possible length among all point sets S C [0, 1)2, where JSI = n. An open problem is to determine the structure of S(n). We show that for any rectangle R contained in [0, 1 F, the number of points in S(n) n R is asymptotic to n times the area of R. One corollary of this result is an 0( n log n) approximation algorithm for the worst-case Euclidean TSP. Analogous results are proved for the minimum spanning tree, minimum-weight matching, and rectilinear Steiner minimum tree. These equidistribution theorems are the first results concerning the structure of worst-case point sets like S(n)

    Learning-Augmented Query Policies for Minimum Spanning Tree with Uncertainty

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    We study how to utilize (possibly erroneous) predictions in a model for computing under uncertainty in which an algorithm can query unknown data. Our aim is to minimize the number of queries needed to solve the minimum spanning tree problem, a fundamental combinatorial optimization problem that has been central also to the research area of explorable uncertainty. For all integral ? ? 2, we present algorithms that are ?-robust and (1+1/?)-consistent, meaning that they use at most ?OPT queries if the predictions are arbitrarily wrong and at most (1+1/?)OPT queries if the predictions are correct, where OPT is the optimal number of queries for the given instance. Moreover, we show that this trade-off is best possible. Furthermore, we argue that a suitably defined hop distance is a useful measure for the amount of prediction error and design algorithms with performance guarantees that degrade smoothly with the hop distance. We also show that the predictions are PAC-learnable in our model. Our results demonstrate that untrusted predictions can circumvent the known lower bound of 2, without any degradation of the worst-case ratio. To obtain our results, we provide new structural insights for the minimum spanning tree problem that might be useful in the context of query-based algorithms regardless of predictions. In particular, we generalize the concept of witness sets - the key to lower-bounding the optimum - by proposing novel global witness set structures and completely new ways of adaptively using those

    Asynchronous Network Formation in Unknown Unbounded Environments

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    In this paper, we study the Online Network Formation Problem (ONFP) for a mobile multi-robot system. Consider a group of robots with a bounded communication range operating in a large open area. One of the robots has a piece of information which has to be propagated to all other robots. What strategy should the robots pursue to disseminate the information to the rest of the robots as quickly as possible? The initial locations of the robots are unknown to each other, therefore the problem must be solved in an online fashion. For this problem, we present an algorithm whose competitive ratio is O(Hmax{M,MH})O(H \cdot \max\{M,\sqrt{M H}\}) for arbitrary robot deployments, where MM is the largest edge length in the Euclidean minimum spanning tree on the initial robot configuration and HH is the height of the tree. We also study the case when the robot initial positions are chosen uniformly at random and improve the ratio to O(M)O(M). Finally, we present simulation results to validate the performance in larger scales and demonstrate our algorithm using three robots in a field experiment

    Neural Algorithmic Reasoning for Combinatorial Optimisation

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    Solving NP-hard/complete combinatorial problems with neural networks is a challenging research area that aims to surpass classical approximate algorithms. The long-term objective is to outperform hand-designed heuristics for NP-hard/complete problems by learning to generate superior solutions solely from training data. The Travelling Salesman Problem (TSP) is a prominent combinatorial optimisation problem often targeted by such approaches. However, current neural-based methods for solving TSP often overlook the inherent "algorithmic" nature of the problem. In contrast, heuristics designed for TSP frequently leverage well-established algorithms, such as those for finding the minimum spanning tree. In this paper, we propose leveraging recent advancements in neural algorithmic reasoning to improve the learning of TSP problems. Specifically, we suggest pre-training our neural model on relevant algorithms before training it on TSP instances. Our results demonstrate that, using this learning setup, we achieve superior performance compared to non-algorithmically informed deep learning models

    Models and Algorithms for Some Covering Problems on Graphs

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    2014 - 2015Several real-life problems as well as problems of theoretical importance within the field of Operations Research are combinatorial in nature. Combinatorial Optimization deals with decision-making problems defined on a discrete space. Out of a finite or countably infinite set of feasible solutions, one has to choose the best one according to an objective function. Many of these problems can be modeled on undirected or directed graphs. Some of the most important problems studied in this area include the Minimum Spanning Tree Problem, the Traveling Salesman Problem, the Vehicle Routing Problem, the Matching Problem, the Maximum Flow Problem. Some combinatorial optimization problems have been modeled on colored (labeled) graphs. The colors can be associated to the vertices as well as to the edges of the graph, depending on the problem. The Minimum Labeling Spanning Tree Problem and the Minimum Labeling Hamiltonian Cycle Problem are two examples of problems defined on edge-colored graphs. Combinatorial optimization problems can be divided into two groups, according to their complexity. The problems that are easy to solve, i.e. problems polynomially solvable, and those that are hard, i.e. for which no polynomial time algorithm exists. Many of the well-known combinatorial optimization problems defined on graphs are hard problems in general. However, if we know more about the structure of the graph, the problems can become more tractable. In some cases, they can even be shown to be polynomial-time solvable. This particularly holds for trees...[edited by Author]XIV n.s

    A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs

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    We consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. For an n×nn\times n 0-1 matrix C,C, let KCK_{C} be the complete weighted graph on the rows of CC where the weight of an edge between two rows is equal to their Hamming distance. Let MWT(C)MWT(C) be the weight of a minimum weight spanning tree of KC.K_{C}. We show that the all-pairs shortest path problem for a directed graph GG on nn vertices with nonnegative real weights and adjacency matrix AGA_G can be solved by a combinatorial randomized algorithm in time O~(n2n+min{MWT(AG),MWT(AGt)})\widetilde{O}(n^{2}\sqrt {n + \min\{MWT(A_G), MWT(A_G^t)\}}) As a corollary, we conclude that the transitive closure of a directed graph GG can be computed by a combinatorial randomized algorithm in the aforementioned time. O~(n2n+min{MWT(AG),MWT(AGt)})\widetilde{O}(n^{2}\sqrt {n + \min\{MWT(A_G), MWT(A_G^t)\}}) We also conclude that the all-pairs shortest path problem for uniform disk graphs, with nonnegative real vertex weights, induced by point sets of bounded density within a unit square can be solved in time O~(n2.75)\widetilde{O}(n^{2.75})
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