Showing papers on "Vertex (graph theory) published in 1999"

Finding the k Shortest Paths

Abstract: We give algorithms for finding the k shortest paths (not required to be simple) connecting a pair of vertices in a digraph. Our algorithms output an implicit representation of these paths in a digraph with n vertices and m edges, in time O(m + n log n + k). We can also find the k shortest paths from a given source s to each vertex in the graph, in total time O(m + n log n + kn). We describe applications to dynamic programming problems including the knapsack problem, sequence alignment, maximum inscribed polygons, and genealogical relationship discovery.

Compact routing with minimum stretch

Abstract: We present the first universal compact routing algorithm with maximum stretch bounded by 3 that uses sublinear space at every vertex. The algorithm uses local routing tables of size O(n2/3log4/3n) and achieves paths that are most 3 times the length of the shortest path distances for all nodes in an arbitrary weighted undirected network. This answers an open question of Gavoille and Gengler who showed that any universal compact routing algorithm with maximum stretch strictly less than 3 must use ?(n) local space at some vertex.

Edge-Coloring Bipartite Multigraphs in $0(E\log D)$ Time

Abstract: Let $V$, $E$, and $D$ denote the cardinality of the vertex set, the cardinality of the edge set, and the maximum degree of a bipartite multigraph $G$. We show that a minimal edge-coloring of $G$ can be computed in $O(E\log D)$ time.

Weights for computing vertex normals from facet normals

Abstract: I propose a new equation to estimate the normal at a vertex of a polygonal approximation to a smooth surface, as a weighted sum of the normals to the facets surrounding the vertex. The equation accounts for the difference in size of these facets by assigning larger weights for smaller facets. When tested on random cubic polynomial surfaces, the equation is superior to other popular weighting methods.

Approximation Algorithms for Maximum Coverage and Max Cut with Given Sizes of Parts

Abstract: In this paper we demonstrate a general method of designing constant-factor approximation algorithms for some discrete optimization problems with cardinality constraints The core of the method is a simple deterministic ("pipage") procedure of rounding of linear relaxations By using the method we design a (1-(1-1/k)k)-approximation algorithm for the maximum coverage problem where k is the maximum size of the subsets that are covered, and a 1/2-approximation algorithm for the maximum cut problem with given sizes of parts in the vertex set bipartition The performance guarantee of the former improves on that of the well-known (1 - e-1)-greedy algorithm due to Cornuejols, Fisher and Nemhauser in each case of bounded k The latter is, to the best of our knowledge, the first constant-factor algorithm for that version of the maximum cut problem

Some properties of the spectrum of graphs

Abstract: Let G be a graph and denote by Q(G)=D(G)+A(G),L(G)=D(G)-A(G) the sum and the difference between the diagonal matrix of vertex degrees and the adjacency matrix of G,respectively. In this paper,some properties of the matrix Q(G)are studied. At the same time,anecessary and sufficient condition for the equality of the spectrum of Q(G) and L(G) is given.

The Spectrum of coupled random matrices

Abstract: The study of the spectrum of coupled random matrices has received rather little attention. To the best of our knowledge, coupled random matrices have been studied, to some extent, by Mehta. In this work, we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled Hermitean random matrices and the equations the associated probabilities satisfy. In particular, the two-Toda lattice, its algebra of symmetries and its vertex operators will play a prominent role in this interaction. Namely, the method is to introduce time parameters, in an artificial way, and to dress up a certain matrix integral with a vertex integral operator, for which we find Virasoro-like differential equations. These methods lead to very simple nonlinear third-order partial differential equations for the joint statistics of the spectra of two coupled Gaussian random matrices. Comment: 56 pages, published version, abstract added in migration

Algebraic analysis of solvable lattice models

Abstract: Background of the problem The spin $1/2$ XXZ model for $\Delta <-1$ The six-vertex model in the anti-ferroelectric regime Solvability and symmetry Correlation functions-physical derivation Level one modules and bosonization Vertex operators Space of states-mathematical picture Traces of vertex operators Correlation functions and form factors The $XXX$ limit $q\rightarrow-1$ Discussions List of formulas

Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously

Abstract: . Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990) and Benjamini and Schramm (1996), we show that the property of having (almost surely) a unique infinite open cluster is increasing in p. Moreover, in the standard coupling of the percolation models for all parameters, a.s. for all p2>p1>pc, each infinite p2-cluster contains an infinite p1-cluster; this yields an extension of Alexander's (1995) “simultaneous uniqueness” theorem. As a corollary, we obtain that the probability θv(p) that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p>pc. All our results extend to quasi-transitive infinite graphs with a unimodular automorphism group.

Axioms for a vertex algebra and the locality of quantum fields

Abstract: The identities satisfied by two-dimensional chiral quantum fields are studied from the point of view of vertex algebras. The Cauchy-Jacobi identity (or the Borcherds identity) for three mutually local fields is proved and consequently a direct proof of Li's theorem on a local system of vertex operators is provided. Several characterizations of vertex algebras are also discussed.

Fast Approximate Graph Partitioning Algorithms

Abstract: We study graph partitioning problems on graphs with edge capacities and vertex weights. The problems of b-balanced cuts and k-balanced partitions are unified into a new problem called minimum capacity $\rho$-separators. A $\rho$-separator is a subset of edges whose removal partitions the vertex set into connected components such that the sum of the vertex weights in each component is at most $\rho$ times the weight of the graph. We present a new and simple O(log n)-approximation algorithm for minimum capacity $\rho$-separators which is based on spreading metrics yielding an O(log n)-approximation algorithm both for b-balanced cuts and k-balanced partitions. In particular, this result improves the previous best known approximation factor for k-balanced partitions in undirected graphs by a factor of O(log k). We enhance these results by presenting a version of the algorithm that obtains an O(log OPT)-approximation factor. The algorithm is based on a technique called spreading metrics that enables us to formulate directly the minimum capacity $\rho$-separator problem as an integer program. We also introduce a generalization called the simultaneous separator problem, where the goal is to find a minimum capacity subset of edges that separates a given collection of subsets simultaneously. We extend our results to directed graphs for values of $\rho \geq 1/2$. We conclude with an efficient algorithm for computing an optimal spreading metric for $\rho$-separators. This yields more efficient algorithms for computing b-balanced cuts than were previously known.

Vertex-reinforced random walk on Z has finite range

Abstract: A stochastic process called vertex-reinforced random walk (VRRW) is defined in Pemantle [Ann. Probab. 16 1229-1241]. We consider this process in the case where the underlying graph is an infinite chain (i.e., process in the case where the underlying graph is an infinite chain (i.e., the one-dimensional integer lattice). We show that the range is almost surely finite, that at least five points are visited infinitely often almost surely and that with positive probability the range contains exactly five points. There are always points visited infinitely often but at a set of times of zero density, and we show that the number of visits to such a point to time n may be asymptotically n α for a dense set of values α ∈ (0, 1). The power law analysis relies on analysis of a related urn model.

Representations of Vertex Operator Algebra V L + for Rank One Lattice L

Abstract: We classify the irreducible modules for the fixed point vertex operator subalgebra V L + of the vertex operator algebra V L associated to a positive definite even lattice of rank 1 under the automorphism lifted from the −1 isometry of L.

Vertex operator solutions to the discrete KP-hierarchy ⁄

Abstract: Vertex operators, which are disguised Darboux maps, transform solutions of the KP equation into new ones. In this paper, we show that the bi-infinite sequence obtained by Darboux transforming an arbitrary KP solution recursively forward and backwards, yields a solution to the discrete KP-hierarchy. The latter is a KP hierarchy where the continuous space x-variable gets replaced by a discrete n-variable. The fact that these sequences satisfy the discrete KP hierarchy is tantamount to certain bilinear relations connecting the consecutive KP solutions in the sequence. At the Grassmannian level, these relations are equivalent to a very simple fact, which is the nesting of the associated infinite-dimensional planes (flag,). The discrete KP hierarchy can thus be viewed as a container for an entire ensemble of vertex or Darboux generated KP solutions. It turns out that many new and old systems lead to such discrete (semi-infinite) solutions, like sequences of soliton solutions, with more and more solitons, sequences of Calogero-Moser systems, having more and more particles, just to mention a few examples; this is developed in [3]. In this paper, as another example, we show that the q-KP hierarchy maps, via a kind of Fourier transform, into the discrete KP hierarchy, enabling us to write down a very large class of solutions to the q-KP hierarchy. This was also reported in a brief note [4].

Characterisation of Graphs which Underlie Regular Maps on Closed Surfaces

Abstract: It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser G e of every edge e is dihedral of order 4 and the stabiliser G v of each vertex v is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incident with v . Such a regular embedding can be realised on an orientable surface if and only if the group G has a subgroup H of index 2 such that H v is the cyclic subgroup of index 2 in G v . An analogous result is proved for orientably-regular embeddings.

On deletion in Delaunay triangulations

Abstract: This paper presents how the space of spheres and shelling may be used to delete a point from a d-dimensional triangu- lation efficiently. In dimension two, if k is the degree of the deleted vertex, the complexity is O( k log k), but we notice that this number only applies to low cost operations, while time consuming computations are only done a linear number of times. This algorithm may be viewed as a variation of Heller’s algorithm [He190, Mid93], which is popular in the geographic information system community. Unfortunately, Heller algorithm is false, as explained in this paper.

On Dominating Sets and Independent Sets of Graphs

Abstract: For a graph G on vertex set V = {1, …, n} let k = (k1, …, kn) be an integral vector such that 1 ≤ ki ≤ di for i ∈ V, where di is the degree of the vertex i in G. A k-dominating set is a set Dk ⊆ V such that every vertex i ∈ VsDk has at least ki neighbours in Dk. The k-domination number γk(G) of G is the cardinality of a smallest k-dominating set of G.For k1 = · · · = kn = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found by N. Alon and J. H. Spencer.If ki = di for i = 1, …, n, then the notion of k-dominating set corresponds to the complement of an independent set. A function fk(p) is defined, and it will be proved that γk(G) = min fk(p), where the minimum is taken over the n-dimensional cube Cn = {p = (p1, …, pn) ∣ pi ∈ R, 0 ≤ pi ≤ 1, i = 1, …, n}. An O(Δ22Δn-algorithm is presented, where Δ is the maximum degree of G, with INPUT: p ∈ Cn and OUTPUT: a k-dominating set Dk of G with ∣Dk∣≤fk(p).

Optimization algorithms for large self-structuring networks

Abstract: As networks grow in scale and heterogeneity, hierarchical organization is inevitable. We present the case for optimal self-organization of network hierarchies. We provide a graph partitioning formulation relevant to networking but different from extant graph partitioning literature. In particular, we require that the resultant partitions be connected, a constraint that changes the character of the problem significantly. We devise a solution that consists of distinct phases for initial partitioning, refinement and post-processing, propose efficient heuristics for each phase, and evaluate them extensively on internetwork-like graphs through simulation. The results suggest that vertex trading techniques, in vogue for a number of decades in graph partitioning, are highly applicable, but multilevel techniques favored by conventional graph partitioning research may be of limited value for internetwork-like graphs. This solution can be implemented in practical networks to automatically generate Internet OSPF areas or ATM PNNI clusters.