Exact colorations of graphs and digraphs

A coloration is an exact regular coloration if whenever two vertices are colored the same they have identically colored neighborhoods. For example, if one of the two vertices that are colored the same is connected to three yellow vertices, two white and red, then the other vertex is as well. Exact regular colorations have been discussed informally in the social network literature. However they have been part of the mathematical literature for some time, though in a different format. We explore this concept in terms of social networks and illustrate some important results taken from the mathematical literature. In addition we show how the concept can be extended to ecological and perfect colorations, and discuss how the CATREGE algorithm can be extended to find the maximal exact regular coloration of a graph.

A generational scheme for partitioning graphs

Graph partitioning divides a graph into several pieces by cutting edges. Very effective heuristic partitioning algorithms have been developed which run in real-time, but it is unknown how good the partitions are since the problem is, in general, NP-complete. This paper reports an evolutionary search algorithm for finding benchmark partitions. Distinctive features are the transmission and modification of whole subdomains (the partitioned units) that act as genes, and the use of a multilevel heuristic algorithm to effect the crossover and mutations. Its effectiveness is demonstrated by improvements on previously established benchmarks.

On Hamilton cycles in locally connected graphs with vertex degree constraints

It is shown that every connected, locally connected graph with the maximum vertex degree Δ(G)=5 and the minimum vertex degree δ(G)3 is fully cycle extendable. For Δ(G)4, all connected, locally connected graphs, including infinite ones, are explicitly described. The Hamilton Cycle problem for locally connected graphs with Δ(G)7 is shown to be NP-complete

Integrating user knowledge with information from parallelisation tools to facilitate the automatic generation of efficient parallel FORTRAN code

User supplied knowledge and interaction is a vital component of a toolkit for producing high quality parallel implementations of scalar FORTRAN numerical code. In this paper we consider the necessary components that such a parallelisation toolkit should possess to provide an effective environment to identify, extract and embed user relevant user knowledge. We also examine to what extent these facilities are available in leading parallelisation tools; in particular we discuss how these issues have been addressed in the development of the user interface of the Computer Aided Parallelisation Tools (CAPTools). The CAPTools environment has been designed to enable user exploration, interaction and insertion of user knowledge to facilitate the automatic generation of very efficient parallel code. A key issue in the user's interaction is control of the volume of information so that the user is focused on only that which is needed. User control over the level and extent of information revealed at any phase is supplied using a wide variety of filters. Another issue is the way in which information is communicated. Dependence analysis and its resulting graphs involve a lot of sophisticated rather abstract concepts unlikely to be familiar to most users of parallelising tools. As such, considerable effort has been made to communicate with the user in terms that they will understand. These features, amongst others, and their use in the parallelisation process are described and their effectiveness discussed.

Models of core/periphery structures

A common but informal notion in social network analysis and other fields is the concept of a core/periphery structure. The intuitive conception entails a dense, cohesive core and a sparse, unconnected periphery. This paper seeks to formalize the intuitive notion of a core/periphery structure and suggests algorithms for detecting this structure, along with statistical tests for testing a priori hypotheses. Different models are presented for different kinds of graphs (directed and undirected, valued and nonvalued). In addition, the close relation of the continuous models developed to certain centrality measures is discussed.

A multilevel algorithm for force-directed graph drawing

We describe a heuristic method for drawing graphs which uses a multilevel technique combined with a force-directed placement algorithm. The multilevel process groups vertices to form clusters, uses the clusters to define a new graph and is repeated until the graph size falls below some threshold. The coarsest graph is then given an initial layout and the layout is successively refined on all the graphs starting with the coarsest and ending with the original. In this way the multilevel algorithm both accelerates and gives a more global quality to the force- directed placement. The algorithm can compute both 2 & 3 dimensional layouts and we demonstrate it on a number of examples ranging from 500 to 225,000 vertices. It is also very fast and can compute a 2D layout of a sparse graph in around 30 seconds for a 10,000 vertex graph to around 10 minutes for the largest graph. This is an order of magnitude faster than recent implementations of force-directed placement algorithms.

A multilevel algorithm for force-directed graph-drawing

We describe a heuristic method for drawing graphs which uses a multilevel framework combined with a force-directed placement algorithm. The multilevel technique matches and coalesces pairs of adjacent vertices to define a new graph and is repeated recursively to create a hierarchy of increasingly coarse graphs, G0, G1, …, GL. The coarsest graph, GL, is then given an initial layout and the layout is refined and extended to all the graphs starting with the coarsest and ending with the original. At each successive change of level, l, the initial layout for Gl is taken from its coarser and smaller child graph, Gl+1, and refined using force-directed placement. In this way the multilevel framework both accelerates and appears to give a more global quality to the drawing. The algorithm can compute both 2 & 3 dimensional layouts and we demonstrate it on examples ranging in size from 10 to 225,000 vertices. It is also very fast and can compute a 2D layout of a sparse graph in around 12 seconds for a 10,000 vertex graph to around 5-7 minutes for the largest graphs. This is an order of magnitude faster than recent implementations of force-directed placement algorithms.

A combined evolutionary search and multilevel optimisation approach to graph-partitioning

The graph-partitioning problem is to divide a graph into several pieces so that the number of vertices in each piece is the same within some defined tolerance and the number of cut edges is minimised. Important applications of the problem arise, for example, in parallel processing where data sets need to be distributed across the memory of a parallel machine. Very effective heuristic algorithms have been developed for this problem which run in real-time, but it is not known how good the partitions are since the problem is, in general, NP-complete. This paper reports an evolutionary search algorithm for finding benchmark partitions. A distinctive feature is the use of a multilevel heuristic algorithm to provide an effective crossover. The technique is tested on several example graphs and it is demonstrated that our method can achieve extremely high quality partitions significantly better than those found by the state-of-the-art graph-partitioning packages.

Single machine scheduling and due date assignment under series-parallel precedence constraints

We consider a single machine due date assignment and scheduling problem of minimizing holding costs with no tardy jobs tinder series parallel and somewhat wider class of precedence constraints as well as the properties of series-parallel graphs.

Visualizing a temporal consistency checker

The notion of time plays a vital and ubiquitous role of a common universal reference. In knowledge-based systems, temporal information is usually represented in terms of a collection of statements, together with the corresponding temporal reference. This paper introduces a visualized consistency checker for temporal reference. It allows expression of both absolute and relative temporal knowledge, and provides visual representation of temporal references in terms of directed and partially weighted graphs. Based on the temporal reference of a given scenario, the visualized checker can deliver a verdict to the user as to whether the scenario is temporally consistent or not, and provide the corresponding analysis / diagnosis.

An analysis of the changes to the mechanical and acoustical properties of paper caused by the damage mechanisms present in the fibre structure

This paper will analyse two of the likely damage mechanisms present in a paper fibre matrix when placed under controlled stress conditions: fibre/fibre bond failure and fibre failure. The failure process associated with each damage mechanism will be presented in detail focusing on the change in mechanical and acoustic properties of the surrounding fibre structure before and after failure. To present this complex process mathematically, geometrically simple fibre arrangements will be chosen based on certain assumptions regarding the structure and strength of paper, to model the damage mechanisms. The fibre structures are then formulated in terms of a hybrid vibro-acoustic model based on a coupled mass/spring system and the pressure wave equation. The model will be presented in detail in the paper. The simulation of the simple fibre structures serves two purposes; it highlights the physical and acoustic differences of each damage mechanism before and after failure, and also shows the differences in the two damage mechanisms when compared with one another. The results of the simulations are given in the form of pressure wave contours, time-frequency graphs and the Continuous Wavelet Transform (CWT) diagrams. The analysis of the results leads to criteria by which the two damage mechanisms can be identified. Using these criteria it was possible to verify the results of the simulations against experimental acoustic data. The models developed in this study are of specific practical interest in the paper-making industry, where acoustic sensors may be used to monitor continuous paper production. The same techniques may be adopted more generally to correlate acoustic signals to damage mechanisms in other fibre-based structures.

A critical examination of node-similarity based graph matching algorithms

In this paper, we shall critically examine a special class of graph matching algorithms that follow the approach of node-similarity measurement. A high-level algorithm framework, namely node-similarity graph matching framework (NSGM framework), is proposed, from which, many existing graph matching algorithms can be subsumed, including the eigen-decomposition method of Umeyama, the polynomial-transformation method of Almohamad, the hubs and authorities method of Kleinberg, and the kronecker product successive projection methods of Wyk, etc. In addition, improved algorithms can be developed from the NSGM framework with respects to the corresponding results in graph theory. As the observation, it is pointed out that, in general, any algorithm which can be subsumed from NSGM framework fails to work well for graphs with non-trivial auto-isomorphism structure.

A robust approach to subsequence matching

In terms of a general time theory which addresses time-elements as typed point-based intervals, a formal characterization of time-series and state-sequences is introduced. Based on this framework, the subsequence matching problem is specially tackled by means of being transferred into bipartite graph matching problem. Then a hybrid similarity model with high tolerance of inversion, crossover and noise is proposed for matching the corresponding bipartite graphs involving both temporal and non-temporal measurements. Experimental results on reconstructed time-series data from UCI KDD Archive demonstrate that such an approach is more effective comparing with the traditional similarity model based algorithms, promising robust techniques for lager time-series databases and real-life applications such as Content-based Video Retrieval (CBVR), etc.