Partitioning Sparse Matrices with Eigenvectors of Graphs

On external-memory MST, SSSP and multi-way planar graph separation

In this paper, we address two longstanding questions about finding good separators in graphs of bounded genus and degree: 1. It is a classical result of Gilbert, Hutchinson, and Tarjan [12] that one can find asymptotically optimal separators on these graphs if he is given both the graph and an embedding of it onto a low genus surface. Does there exist a simple, efficient algorithm to find these separators given only the graph and not the embedding? 2. In practice, spectral partitioning heuristics work extremely well on these graphs. Is there a theoretical reason why this should be the case.We resolve these two questions by showing that a simple spectral algorithm finds separators of cut ratio O(√g/n) and vertex bisectors of size O(√gn) in these graphs, both of which are optimal. As our main technical lemma, we prove an O(g/n) bound on the second smallest eigenvalue of the Laplacian of such graphs and show that this is tight, thereby resolving a conjecture of Spielman and Teng. While this lemma is essentially combinatorial in nature, its proof comes from continuous mathematics, drawing on the theory of circle packings and the geometry of compact Riemann surfaces.

We exhibit a close relationship between two topics in computational complexity. One topic is the analysis of storage requirements for nondeterministic computations. The corresponding mathematical model is a well known black-white pebble game on directed acyclic graphs. The other topic is the search for small separators of undirected graphs. We model a dynamic version of the concept of a separator with a vertex separator game. This game is closely related to graph layout and searching problems. We show that instances of the black-white pebble game and the vertex separator game can easily be transformed into each other. As an application of this result both games are shown to be NP-complete.

The structure of this survey is the following: at first we repeat in Part I the first 20 years old proof of the strong Elliptic law for random matrices Then in Part II we prove the strong Elliptic law for random matrices Ξ n of the general form, i.e. when their diagonal entries have nonzero expectations, and when we do not require the existence of the probability densities of the entries of random matrices, but instead of Lyapunov condition we require the boundedness of the absolute moments of the entries of the order 4 + δ , δ &gt; 0. In this case the Elliptic law means that the support of the accompanying spectral density of eigenvalues looks like the picture of several galaxies made by telescope. If the distances between the centers of these galaxies are large enough we have several almost elliptical galaxies. We also present the sand clock law density for the sum of diagonal complex matrix and the symmetric random matrix belonging the domain of attraction of the Semicircular law. After this we prove in the Part III the weak Elliptic law, when all entries can have nonzero expectations. At the end of survey we give the proof of the Elliptic law for the Unitary random matrices from the so called Class N 12 of random Unitary matrices and give the over turned stool law density for the sum of diagonal complex matrix A n and where Ξ n belongs to the domain of attraction of the Circular law. We also establish in Part III for Elliptic Law at the first time the convergency rate and the Central Limit Theorem. These statements are based on the VICTORIA -transform of random matrix which is the abbreviation of the following words: Very Important Computational Transformation Of Random Independent Arrays. We follow the main strategy of the theory of limit theorems of the probability theory, i.e. we try to solve the problem of description of all limits of normalized spectral functions where λ k ( A n Ξ n B n + C n ) are eigenvalues of the matrix A n Ξ n B n + C n , A n , B n , and C n are nonrandom matrices, under general (as only possible) conditions on the entries of random matrices Ξ n , χ is the indicator function. We emphasize that the spectral theory of Hermitian random matrices is rather profound. For example, in 1975 in [3] V. Girko proved the general stochastic canonical equation for ACE (Asymptotically Constant Entries)-symmetric matrices: Assume that for any n , the random entries of a symmetric matrix are independent and they are asymptotically constant entries (ACE), i.e., for any ε &gt; 0, and τ &gt; 0 is an arbitrary constant, and that, for every 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1, where the symbol denotes the weak convergence of distribution when n → ∞, is a nondecreasing function with bounded variation in z and continuous in u and v in the domain 0 ≤ u , v ≤ 1. Then, with probability one, for almost all x , where λ k (Ξ n × n ) are eigenvalues, F ( x ) is a distribution function whose Stieltjes transform satisfies the relation G α ( x , y , t ), as a function of x , is a distribution function satisfying the regularized stochastic canonical equation K 3 [3, 23] at the points x of continuity, is a random real functional whose Laplace transform of one-dimensional distribution is equal to The integrand <jats:inli

In this paper, the third eigenvalue of the Laplacian matrix is used to provide a lower bound on the minimum cutsize. This result has algorithmic implications that are exploited in this paper. Besides, combinatorial properties of certain configurations of a graph partition which are related to the minimality of a cut are investigated. It is shown that such configurations are related to the third eigenvector of the Laplacian matrix. It is well known that the second eigenvector encodes structural information, and that can be used to approximate a minimum bisection. In this paper, it is shown that the third eigenvector carries structural information as well. Then a new spectral bisection algorithm using both eigenvectors is provided. The new algorithm is guaranteed to return a cut that is smaller or equal to the one returned by the classic spectral bisection. Also, a spectral algorithm that can refine a given partition and produce a smaller cut is provided.

A k-ratio edge separator is a set of edges which separates a weighted graph into two disconnected sets of components neither of which contains more than k-1/k of the original graph's weight. An optimal quotient separator is an edge separator where the size of the separator (i.e., the number of edges) divided by the weight of the smaller set of components is minimized. An optimal quotient k-ratio separator is an edge separator where the size of the separator (i.e., the number of edges) divided by the smaller of either 1/k of the total weight or the weight of the smaller set of components is minimized. In this paper we present an algorithm that finds the optimal quotient k-ratio separator for any k ≥ 3. We use the optimal quotient algorithm to obtain approximation algorithms for finding optimal k-ratio edge separators for any k ≥ 3. Given a planar graph with a size OPT k-ratio separator, we describe an algorithm which a finds k-ratio separator which costs less than O(OPT log n). More importantly the algorithm finds ck-ratio separators (for any c ≫ 1) which cost less than C(c)OPT, where C(c) depends only on c.

In paper the generalized real eigenvalue and fuzzy eigenvector of a crisp real symmetric matrix with respect to another real symmetric matrix is studied. The original generalized fuzzy eigen problem is extended into a crisp generalized eigen problem of a real symmetric matrix with high orders using the arithmetic operation of LR fuzzy matrix and vector. Two cases are analysed: (a) the unknown eigenvalue λ is a non negative real number; (b) the unknown eigenvalue λ is a negative real number. Two computing models are established and an algorithm for finding the generalized fuzzy eigenvector of a real symmetric matrix is derived. Moreover, a sufficient condition for the existence of a strong generalized fuzzy eigenvector is given. Some numerical examples are shown to illustrated our proposed method.

In this paper, we propose a depth-first search (DFS) algorithm for finding maximum matching in general graphs. Unlike blossom shrinking algorithms, which store all possible alternative alternating-paths in the super-vertices shrinking from blossoms, the newly proposed algorithm does not involve blossom shrinking. The basic idea is to deflect the alternating path when facing blossoms. The algorithm maintains detour information in an auxiliary stack to minimize the redundant data structures. A benefit of our technique is to avoid spending the time on shrinking and expanding blossoms. This DFS algorithm can determine a maximum matching of a general graph with m edges and n vertices in 𝑂(𝑚𝑛) time with space complexity 𝑂(𝑛).

A heuristic graph partitioning scheme is presented to determine effective node separators for undirected graphs. An initial separator is first obtained from the minimum degree ordering, an algorithm designed originally to produce fill-reducing orderings for sparse matrices. The separator is then improved by an iterative strategy based on some known results from bipartite graph matching. This gives an overall practical scheme in partitioning graphs. Experimental results are provided to demonstrate the effectiveness of this heuristic algorithm on graphs arising from sparse matrix applications.

Attacks launched over the Internet have become a pressing problem. Attackers make use of a variety of techniques to anonymize their traffic, in order to escape detection and prosecution. Despite much research on attack attribution, there has been little work on optimizing the number and placement of monitoring points for identifying the source of attacks with minimum ambiguity. This paper proposes such a method. The approach is based on the concept of graph separators. A separator partitions a network, such that the size of the separator is the number of monitors needed, and the size of a partition is the ambiguity in isolating the specific source of an attack. To achieve a desired degree of ambiguity, a good separator for the Internet is sought. Both vertex and edge separator heuristics are presented, which greedily select vertices of highest/lowest degree as monitors. The methods are evaluated for the Internet autonomous system (AS) topology. Experimental results show that the vertex separator heuristic requires just 5% of the ASes to be monitored to identify the source of an attack with little ambiguity. If only those links actually used for routing to a specific destination are considered, use of an edge separator requires 30% of the links to be monitored to achieve similar results. The results can be further improved if it is known that ASes have unequal probabilities of being the source of an attack.

The set of Fortran subroutines discussed an implementation of the algorithm for finding the eigenvectors, x, and eigenvalues, lambda, such that Ax = lambdax, where A is a real skew-symmetric matrix or a real tridiagonal symmetric matrix with a constant diagonal. The algorithm uses only orthogonal similarity transformations and is believed to be the most efficient procedure available for computing all the eigenvalues or the complete eigensystem for the indicated classes of matrices. The three subroutines of the algorithm and their functions are described as follows: TRIZD--a subroutine that transforms an arbitrary real skew-symmetric matrix to skew-symmetric tridiagonal form by using orthogonal similarity transformations; IMZD--a subroutine that computes the eigenvalues and, optionally, the eigenvectors of a symmetric tridiagonal matrix with zeros on the diagonal or of a skew-symmetric tridiagonal matrix; TBAKZD--a subroutine that computes the eigenvectors of an arbitrary real skew-symmetric matrix by back-transforming the eigenvectors of the corresponding skew-symmetric tridiagonal matrix determined by TRIZD. The subroutines TRIZD, IMZD, and TBAKZD have been tested extensively on an IBM 360/91 computer using double precision arithmetic. Complete subroutine listings are available. (RWR)

The algorithm is a new approach for solving eigenvalue problem which is based on the P (p, q, w) rotation transform; it is the effective way for solving all the eigenvalues and eigenvectors of the general matrix at present, and it is applicable to both cases of the symmetric matrix and non-symmetric matrix. The method uses Householder matrix, first matrix A is transformed into a symmetric tridiagonal matrix, and then use computer simulation for QR algorithm to achieve the eigenvalues of symmetric triangular matrix. QR Algorithm is one of the most effective ways for solving all the eigenvalues and eigenvectors of the small and medium sized matrix at present, and it has good stability and rapid convergence speed.

Matchability and [formula omitted]-maximal matchings

To this day, there are still some countries where the exchange of kidneys between multiple incompatible patient-donor pairs is restricted by law. Typically, legal regulations in this context are put in place to prohibit coercion and manipulation in order to prevent a market for organ trade. Yet, in countries where kidney exchange is practiced, existing platforms to facilitate such exchanges generally lack sufficient privacy mechanisms. In this paper, we propose a privacy-preserving protocol for kidney exchange that not only addresses the privacy problem of existing platforms but also is geared to lead the way in overcoming legal issues in those countries where kidney exchange is still not practiced. In our approach, we use the concept of secret sharing to distribute the medical data of patients and donors among a set of computing peers in a privacy-preserving fashion. These computing peers then execute our new Secure Multi-Party Computation (SMPC) protocol among each other to determine an optimal set of kidney exchanges. As part of our new protocol, we devise a privacy-preserving solution to the maximum matching problem on general graphs. We have implemented the protocol in the SMPC benchmarking framework MP-SPDZ and provide a comprehensive performance evaluation. Furthermore, we analyze the practicality of our protocol when used in a dynamic setting (where patients and donors arrive and depart over time) based on a data set from the United Network for Organ Sharing.

An external-memory depth-first search algorithm for general grid graphs