74,680 research outputs found
Computing Minimum Spanning Trees with Uncertainty
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 of the graph only a set , called an uncertainty area, that
contains the actual edge weight is known. The algorithm can `update'
to obtain the edge weight . The task is to output the edge set of
a minimum spanning tree after a minimum number of updates. An algorithm is
-update competitive if it makes at most times as many updates as the
optimum. We present a 2-update competitive algorithm if all areas are
open or trivial, which is the best possible among deterministic algorithms. The
condition on the areas 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
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
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
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
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
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 for arbitrary robot deployments, where is the
largest edge length in the Euclidean minimum spanning tree on the initial robot
configuration and 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 . 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
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
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
We consider the problem of computing all-pairs shortest paths in a directed
graph with real weights assigned to vertices.
For an 0-1 matrix let be the complete weighted graph
on the rows of where the weight of an edge between two rows is equal to
their Hamming distance. Let be the weight of a minimum weight spanning
tree of
We show that the all-pairs shortest path problem for a directed graph on
vertices with nonnegative real weights and adjacency matrix can be
solved by a combinatorial randomized algorithm in time
As a corollary, we conclude that the transitive closure of a directed graph
can be computed by a combinatorial randomized algorithm in the
aforementioned time.
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
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