The Optimal Graph Whose Least Eigenvalue is Minimal among All Graphs via 1-2 Adjacency Matrix

All graphs under consideration are finite, simple, connected, and undirected. Adjacency matrix of a graph G is 0,1 matrix A = a i j = 0 , i f v i = v j o r d v i , v j ≥ 2 1 , i f d v i , v j = 1. . Here in this paper, we discussed new type of adjacency matrix known by 1-2 adjacency matrix defined as A 1,2 G = a i j = 0 , i f v i = v j o r d v i , v j ≥ 3 1 , i f d v i , v j = 2 , from eigenvalues of the graph, we mean eigenvalues of the 1-2 adjacency matrix. Let T n c be the set of the complement of trees of order n . In this paper, we characterized a unique graph whose least eigenvalue is minimal among all the graphs in T n c .

MP-RRT#: a Model Predictive Sampling-based Motion Planning Algorithm for Unmanned Aircraft Systems

This paper introduces a kinodynamic motion planning algorithm for Unmanned Aircraft Systems (UAS), called MP-RRT#. MP-RRT# joins the potentialities of RRT# with a strategy based on Model Predictive Control to efficiently solve motion planning problems under differential constraints. Similar to other RRT-based algorithms, MP-RRT# explores the map constructing an asymptotically optimal graph. In each iteration the graph is extended with a new vertex in the reference state of the UAS. Then, a forward simulation is performed using a Model Predictive Control strategy to evaluate the motion between two adjacent vertices, and a trajectory in the state space is computed. As a result, the MP-RRT# algorithm eventually generates a feasible trajectory for the UAS satisfying dynamic constraints. Simulation results obtained with a simulated drone controlled with the PX4 autopilot corroborate the validity of the MP-RRT# approach.

Optimal graph edge weights driven nlms with multi-layer residual compensation

Non-local Means (NLMs) play essential roles in image denoising, restoration, inpainting, etc., due to its simple theory but effective performance. However, when the noise increases, the denoising accuracy of NLMs decreases significantly. This paper further develop the NLMs-based denoising method to remove noise with less loss of image details. It is realized by embedding an optimal graph edge weights driven NLMs kernel into a multi-layer residual compensation framework. Unlike the patch similarity-based weights in the traditional NLMs filters, the edge weights derived from the optimal graph Laplacian regularization consider (1) the distance between the target pixel and the candidate pixel, (2) the local gradient and (3) the patch similarity. After defining the weights, the graph-based NLMs kernel is then put into a multi-layer framework. The corresponding primal and residual terms at each layer are finally fused with learned weights to recover the image. Experimental results show that our method is effective and robust, especially for piecewise smooth images.

Tracing Topic Transitions with Temporal Graph Clusters

Twitter serves as a data source for many Natural Language Processing (NLP) tasks. It can be challenging to identify topics on Twitter due to continuous updating data stream. In this paper, we present an unsupervised graph based framework to identify the evolution of sub-topics within two weeks of real-world Twitter data. We first employ a Markov Clustering Algorithm (MCL) with a node removal method to identify optimal graph clusters from temporal Graph-of-Words (GoW). Subsequently, we model the clustering transitions between the temporal graphs to identify the topic evolution. Finally, the transition flows generated from both computational approach and human annotations are compared to ensure the validity of our framework.

Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

We study algorithmic properties of the graph class $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e , that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. It appears that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e . More precisely, we identify a large class of optimization problems on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e solvable in time $$2^{{\mathcal{O}}(\sqrt{k}\log k)}\cdot n^{{\mathcal{O}}(1)}$$ 2 O ( k log k ) · n O ( 1 ) . Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. Besides subexponential time algorithms, the class of $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e , namely, $${\textsc {Interval}}{-ke}$$ I N T E R V A L - k e and $${\textsc {Split}}{-ke}$$ S P L I T - k e graphs.

Optimal Graph Edge Weights Driven NLMs with Multi-layer Residual Compensation

Non-Local Means (NLMs) play important roles in image denoising, restoration, inpainting etc. due to its simple theory but effective performance. In this paper, in order to better remove noise without loss of image details, we further develop the NLMs based denoising method. It is realized by introducing a graph Laplacian regularization based weighting model and a multi-layer residual compensation strategy. Unlike the patch similarity based weights in classic NLMs filters, the graph Laplacian regularization defines the weights by considering 1) the distance between target pixel and the candidate pixel, 2) the local gradient and 3) the patch similarity. The proposed NLMs filter performs in a multi-layer framework to better remove the noise and smooth the result. The corresponding residuals at each layer are finally combined with the smooth image with learned weights to recover the image details. Experimental results show that our method is effective and robust, especially for piecewise-smooth images.

Near-Optimal Graph Signal Sampling by Pareto Optimization

In this paper, we focus on the bandlimited graph signal sampling problem. To sample graph signals, we need to find small-sized subset of nodes with the minimal optimal reconstruction error. We formulate this problem as a subset selection problem, and propose an efficient Pareto Optimization for Graph Signal Sampling (POGSS) algorithm. Since the evaluation of the objective function is very time-consuming, a novel acceleration algorithm is proposed in this paper as well, which accelerates the evaluation of any solution. Theoretical analysis shows that POGSS finds the desired solution in quadratic time while guaranteeing nearly the best known approximation bound. Empirical studies on both Erdos-Renyi graphs and Gaussian graphs demonstrate that our method outperforms the state-of-the-art greedy algorithms.