Optimizing Partitions of Percolating Graphs

The partitioning of random graphs is investigated numerically using \simulated annealing" and \extremal optimization." While generally an NP-hard problem, it is shown that the optimization of the graph partitions is particularly diicult for sparse graphs with average connectivities near the percolation threshold. At this threshold, the relative error of \simulated annealing" is found to diverge in the thermodynamic limit. On the other hand, \extremal optimization," a new general purpose method based on self-organized criticality, produces near-optimal partitions with bounded error at any low connectivity at a comparable computational cost. The partitioning of graphs is generally an NP-hard optimization problem with many practical applications such as VLSI design (1) and load-balancing between parallel processors (2). In physics it is most closely related to nding ground states of spin-glasses, which can be mapped into a graph partitioning problem (3). The graph (bi-)partitioning problem is easy to formulate: Take N points where N is an even number, let any pair of two points be connected by an edge with a probability p, divide the points into two sets of equal size N=2 such that the number of edges connecting both sets, m, is minimal: m = m opt. The global constraint of an equal division of the points between the sets places this problem generally among the hardest problems in combinatorial optimization, requiring a computational eeort that would grow faster than any power of N to determine the exact solution with certainty. Fast heuristic methods, such as Kernighan-Lin (4), are moderately successful in nding some of a huge number of near-optimal conngurations. In this paper we will focus instead on two more generally applicable optimization methods: simulated annealing (SA) (3) and extremal optimization (EO) (5). The graph partitioning problem depends on the probability p with which any two points in the system are connected. Thus, p determines the total number of