N Increases. in the Region 0:40 < <

1, the KK heuristic performs poorly as n increases. This is despite the fact that these problems have perfect partitions that can often be found with little search. 1 1e+10 1e+20-40-20 0 20 40 60 80 100 "n=6" "n=12" "n=18" "n=24" Figure 9. Average size of KK partition, hKKi against. Performance guarantees for optimization procedures are often given as a ratio of the optimal value. Since the optimal partition can have a diierence of zero, such a ratio would be undeened for perfect partitions. We therefore make a \mean-eld" approximation that, h KK opt i h KK i hopti : The mean optimal partition size, hopti is at least 1/2 so the performance ratio is always deened. In Figure 10, we plot the average performance ratio for the KK heuristic against again for n = 6 to 24 and log 2 (l) from 0 to 2n. The maximum performance ratio grows approximately as a simple exponential in n.