The Peculiar Phase Structure of Random Graph Bisection

The mincut graph bisection problem involves partitioning the n vertices of a graph into disjoint subsets, each containing exactly n/2 vertices, while minimizing the number of "cut" edges with an endpoint in each subset. When considered over sparse random graphs, the phase structure of the graph bisection problem displays certain familiar properties, but also some surprises. It is known that when the mean degree is below the critical value of 2 log 2, the cutsize is zero with high probability. We study how the minimum cutsize increases with mean degree above this critical threshold, finding a new analytical upper bound that improves considerably upon previous bounds. Combined with recent results on expander graphs, our bound suggests the unusual scenario that random graph bisection is replica symmetric up to and beyond the critical threshold, with a replica symmetry breaking transition possibly taking place above the threshold. An intriguing algorithmic consequence is that although the problem is NP-hard, we can find near-optimal cutsizes (whose ratio to the optimal value approaches 1 asymptotically) in polynomial time for typical instances near the phase transition.Comment: substantially revised section 2, changed figures 3, 4 and 6, made minor stylistic changes and added reference

Bipartição de redes de pequeno mundo

We studied the bipartitioning of small-world networks. We generated small-world networks from a square lattice via a modified Watts- Strogatz algorithm. We compared several partitioning algorithms, such as Monte Carlo with Kawasaki dynamics and Simulated Annealing, Extremal Optimization and Multilevel K-way partitioning. We obtained the critical percolation values of the mean degree and the threshold partition values for several networks in the continuum between a square lattice and a random network. We obtained the exponents of the minimum partition cost as a function of the mean degree in the vicinity of the bipartition threshold, as well as the exponent of finite size scaling. To the best of our knowledge, this is the first work to tackle this issue. We observed that all small-world networks have the same exponents, different from those of the square lattice, regardless of the number of modified edges. The values for the exponents of the random network are in accordance with previous results.Neste estudo, investigámos a bipartição de redes de pequeno mundo. Utilizámos um modelo de Watts-Strogatz modificado para a geração de redes de pequeno mundo a partir de redes quadradas. Comparámos vários algoritmos de partição, tais como Monte Carlo com dinâmica Kawasaki e Simulated Annealing, Extremal Optimization e Multilevel K-way Based Partitioning. Obtivemos os valores críticos do grau médio das redes na transição de percolação e os valores limite na transição de partição para redes ao longo do espetro entre uma rede quadrada e uma rede aleatória. Obtivemos os expoentes de variação do custo de partição mínimo em função da grau médio da rede na proximidade do valor limite de partição, assim como o expoente de escalonamente em função do tamanho da rede. Este é o primeiro trabalho a estudar este problema, tanto quanto sabemos. Observou-se que as redes modificadas apresentam os mesmos expoente, independemente do número de arestas modificadas, enquanto que a rede quadrada tem um comportamento distinto de todas as redes de pequeno mundo. Os valores dos expoentes para a rede aleatória estão em concordância com resultados prévios.Mestrado em Físic