Extremal Optimization of Graph Partitioning at the Percolation Threshold

The benefits of a recently proposed method to approximate hard optimization problems are demonstrated on the graph partitioning problem. The performance of this new method, called Extremal Optimization, is compared to Simulated Annealing in extensive numerical simulations. While generally a complex (NP-hard) problem, the optimization of the graph partitions is particularly difficult for sparse graphs with average connectivities near the percolation threshold. At this threshold, the relative error of Simulated Annealing for large graphs is found to diverge relative to Extremal Optimization at equalized runtime. On the other hand, Extremal Optimization, based on the extremal dynamics of self-organized critical systems, reproduces known results about optimal partitions at this critical point quite well.Comment: 7 pages, RevTex, 9 ps-figures included, as to appear in Journal of Physics

On the extremal properties of the average eccentricity

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity $ecc (G)$ of a graph $G$ is the mean value of eccentricities of all vertices of $G$. The average eccentricity is deeply connected with a topological descriptor called the eccentric connectivity index, defined as a sum of products of vertex degrees and eccentricities. In this paper we analyze extremal properties of the average eccentricity, introducing two graph transformations that increase or decrease $ecc (G)$. Furthermore, we resolve four conjectures, obtained by the system AutoGraphiX, about the average eccentricity and other graph parameters (the clique number, the Randi\' c index and the independence number), refute one AutoGraphiX conjecture about the average eccentricity and the minimum vertex degree and correct one AutoGraphiX conjecture about the domination number.Comment: 15 pages, 3 figure

Extremal Optimization for Graph Partitioning

Extremal optimization is a new general-purpose method for approximating solutions to hard optimization problems. We study the method in detail by way of the NP-hard graph partitioning problem. We discuss the scaling behavior of extremal optimization, focusing on the convergence of the average run as a function of runtime and system size. The method has a single free parameter, which we determine numerically and justify using a simple argument. Our numerical results demonstrate that on random graphs, extremal optimization maintains consistent accuracy for increasing system sizes, with an approximation error decreasing over runtime roughly as a power law t^(-0.4). On geometrically structured graphs, the scaling of results from the average run suggests that these are far from optimal, with large fluctuations between individual trials. But when only the best runs are considered, results consistent with theoretical arguments are recovered.Comment: 34 pages, RevTex4, 1 table and 20 ps-figures included, related papers available at http://www.physics.emory.edu/faculty/boettcher