Eigensolution of Laplacian matrices for graph partitioning and domain decomposition

The problem of computing a small vertex separator in a graph arises in the context of computing a good ordering for the parallel factorization of sparse, symmetric matrices. An algebraic approach for computing vertex separators is considered in this paper. It is, shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph. The Laplacian eigenvectors of grid graphs can be computed from Kronecker products involving the eigenvectors of path graphs, and these eigenvectors can be used to compute good separators in grid graphs. A heuristic algorithm is designed to compute a vertex separator in a general graph by first computing an edge separator in the graph from an eigenvector of the Laplacian matrix, and then using a maximum matching in a subgraph to compute the vertex separator. Results on the quality of the separators computed by the spectral algorithm are presented, and these are compared with separators obtained from other algorith...

For network graphs, numerous graph features are intimately linked to eigenvalues of the Laplacian matrix, such as connectivity and diameter. Thus, it is very important to solve eigenvalues of the Laplacian matrix for graphs. Similarly, for higher-order networks, eigenvalues of combinatorial Laplacian matrices are also important for invariants of graphs. However, for large-scale networks, it is difficult to calculate eigenvalues of the Laplacian matrix directly because it is either very difficult to obtain the whole network structure or requires a lot of computing resources. Therefore, this article makes the following contributions. Firstly, this paper proposes a random walk approach for estimating the bounds of the greatest eigenvalues of Laplacian matrices for large-scale networks. Considering the relationship between the spectral moments of the adjacency matrix and the closed paths in the network, we utilize the relationship between the adjacency matrix and the Laplacian matrix to establish the relationship between the Laplacian matrix and the closed paths. Then, we employ equiprobable random walks to sample the large graph to obtain the small graph. Through algebraic topology knowledge, we obtain the bounds of the largest eigenvalue of the Laplacian matrix of the large graph by using Laplacian spectral moments of the small graph. Secondly, for high-order networks, this paper proposes a method based on random walks and graph transformations. The graph transformation we propose mainly converts graphs with second-order simplices into ordinary weighted graphs, thereby transforming the problem of solving the spectral moments of the second-order combined Laplacian matrix into solving the spectral moments of the adjacency matrix. Then, we use the aforementioned random walk method to solve bounds of the greatest eigenvalue of the second-order combinatorial Laplacian matrix. Finally, by comparing the proposed method with existing algorithms in synthetic and real networks, its accuracy and superiority are demonstrated.

OF THE DISSERTATION Two applications of combinatorial optimization by Matthew R. Oster Dissertation Director: Professor Jonathan Eckstein This thesis presents two applications of combinatorial optimization. The first part contains a detailed description of a conference scheduling problem. We model the problem as a symmetric clustering problem, or a variant of minimum k-partition we call capacitated k-partition. This problem is proved to be NP-hard to solve to optimality, and further, unless P = NP, no constant factor polynomial-time approximation algorithm exists. We also propose a branch-and-cut algorithm with semidefinite programming relaxations enhanced with polyhedral cuts found at each tree node. Many cutting planes are demonstrated to be satisfied, or provably close to being satisfied, by semidefinite matrices in the variable space [−1/(k − 1), 1], which is in contrast to basic linear programming relaxations. Our algorithm also relies on a novel heuristic strategy when attempting to generate feasible solutions at every tree node. We test an implementation of our algorithm on random k-partition instances as well as a particular conference data set which comes from the 13th Annual INFORMS Computing Society Conference and was solved to within 0.85% of optimum in under 4 hours. The results here are promising and provide a starting point for future projects. In the second part, we describe a project called the Boat Allocation Module, where a team comprised of United States Coast Guard (USCG) analysts, and Command, Control, and Interoperability Center for Advanced Data Analysis researchers worked

Automatic detection of salient image regions is a useful tool with applications in intelligent camera control, virtual cinematography, video summarization and editing, evaluation of viewer preferences, and many others. This paper presents an effective method for detecting potentially salient foreground regions. Salient regions are identified by eigenvalue analysis of a graph Laplacian that is defined over the color similarity of image superpixels, under the assumption that the majority of pixels on the image boundary show non-salient background. In contrast to previous methods based on graph-cuts or graph partitioning, our method provides continuously-valued saliency estimates with complementary properties to recently proposed color contrast-based approaches. Moreover, exploiting discriminative properties of the Fiedler vector, we devise an SVM-based classifier that allows us to determine whether an image contains any salient objects at all, a problem that has been largely neglected in previous works. We also describe how the per-frame saliency detection can be extended to improve its spatiotemporal coherence when computed on video sequences. Extensive evaluation on several datasets demonstrates and validates the state-of-the-art performance of the proposed method.

Eigenvalues and eigenvectors have many applications in structural mechanics and combinatorial optimization. In this paper, a set of matrices of special forms is studied for which the calculation of eigenvalues can be performed much easier than with the existing general methods. First tri-diagonal matrices are presented and then the relationships for calculating their eigenvalues are extended to the evaluation of the eigenvalues of block tri-diagonal matrices. Block penta-diagonal matrices are also studied in this paper. The eigensolution of different problems of structural mechanics is performed to show the simplicity of using the present formulations.

As a well‐structured and costly activity that pervades industries in both the public and private sector, vehicle fleet management would appear to be a splendid candidate for model‐based planning and optimization. And yet, until recently the combinatorial intricacies of vehicle routing and of vehicle scheduling have precluded the widespread use of optimization (exact) methods for this problem class. Our discussion in this paper identifies the extent and nature of these problem complexities and draws contrasts with other applications of combinatorial optimization. It also summarizes a number of successful uses of optimization for vehicle fleet planning and highlights potentially fruitful avenues for algorithmic development. In particular, we describe several alternative models and novel algorithms for the vehicle routing problem, show how various modeling approaches for this problem are intimately related, and illustrate the interplay between model formulations and the algorithms that they suggest. This discussion shows that prospects for applying exact methods, possibly in conjunction with heuristics, are far from fully realized and points to vehicle fleet planning as a tempting target of opportunity for further investigation.

Finding the largest suborder of fixed width in a partially ordered set is an interesting combinatorial problem with applications in combinatorial optimization and scheduling. We present a polynomial time solution for this problem by transforming it into a minimum cost network flow problem in an appropriate auxiliary network.

Combinatorial optimization problems are becoming more and more important as its applications take place in many aspects, especially in logistics which planning and management of warehouse are mostly based on. Using exhaustive search for a precise solution is likely not preferable, since the complexity and execution time increases as factorial of input. So people attempt to look for a method which could provide a reasonably good solution with fast response, which heuristic and metaheuristic approaches are superior. In this paper, we will discuss briefly about some common heuristic and metaheuristic ones, as well as their principles and how they are applied to seek the optimal routes in a predefined graph. Then we come to a realistic application in logistics that AGVs have to allocate the best routes and order to achieve the tasks given, by simulation in MATLAB GUI program.

This paper deals with algorithms for positive semidefinite matrix factorization (PSDMF). PSDMF is a recently-proposed extension of nonnegative matrix factorization with applications in combinatorial optimization, among others. In this paper, we focus on improving the local convergence of an alternating block gradient (ABC) method for PSDMF in a noise-free setting by replacing the quadratic objective function with the Poisson log-likelihood. This idea is based on truncated Wirtinger flow (TWF), a phase retrieval (PR) method that trims outliers in the gradient and thus regularizes it. Our motivation is a recent result linking PR with PSDMF. Our numerical experiments validate that the numerical benefits of TWF may carry over to PSDMF despite the more challenging setting, when initialized within its region of convergence. We then extend TWF from PR to affine rank minimization (ARM), and show that although the outliers are no longer an issue in the ARM setting, PSDMF with the new objective function may still achieves a smaller error for the same number of iterations. In a broader view, our results indicate that a proper choice of objective function may enhance convergence of matrix (or tensor) factorization methods.

The comparative performance of Integer Programming (IP) and Constraint Logic Programming (CLP) is explored by examining a number of models for four different combinatorial optimization applications. Computational results show contrasting behavior for the two approaches, and an analysis of performance with respect to problem and model characteristics is presented. The analysis shows that tightness of formulation is of great benefit to CLP where effective search reduction results in problems that can be solved quickly. In IP, if the linear feasible region does not identify the corresponding integer polytope, the problem may be difficult to solve. The paper identifies other characteristics of model behavior and concludes by examining ways in which IP and CLP may be incorporated within hybrid solvers.

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A continuous quadratic programming formulation is given for min-cut graph partitioning problems. In these problems, we partition the vertices of a graph into a collection of disjoint sets satisfying specified size constraints, while minimizing the sum of weights of edges connecting vertices in different sets. An optimal solution is related to an eigenvector (Fiedler vector) corresponding to the second smallest eigenvalue of the graph's Laplacian. Necessary and sufficient conditions characterizing local minima of the quadratic program are given. The effect of diagonal perturbations on the number of local minimizers is investigated using a test problem from the literature.

The m‐way graph partitioning problem is of central importance in combinatorial optimization. It has many important applications in fields such as VLSI circuit design, task allocation in distributed computing systems, and network partitioning. In this paper, we propose an efficient genetic algorithm to solve this problem. The proposed method searches a large solution space and finds the best possible solution by adjusting the intensification and diversification automatically during the optimization process. The proposed method is tested on a large number of instances and compared with some existing algorithms. The experimental results show that the proposed algorithm is superior to its competitors in terms of computation time and solution quality. © 2011 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Networks are continuously growing in complexity, which creates challenges for determining their most important characteristics. While analytical bounds are often too conservative, the computational effort of algorithmic approaches does not scale well with network size. This work uses Cartesian Genetic Programming for symbolic regression to evolve mathematical equations that relate network properties directly to the eigenvalues of network adjacency and Laplacian matrices. In particular, we show that these eigenvalues are powerful features to evolve approximate equations for the network diameter and the isoperimetric number, which are hard to compute algorithmically. Our experiments indicate a good performance of the evolved equations for several real-world networks and we demonstrate how the generalization power can be influenced by the selection of training networks and feature sets.

Summary of Volume 3: Applications of Combinatorial Optimization