Extremal optimization: heuristics via coevolutionary avalanches

gates on a single integrated circuit, so you have to partition the gates between separate circuits. Let's assume that you are forced to place exactly n/2 gates each on two integrated circuits. The connections between the gates across the partition are slow, energy consuming, and heat producing, while the cost associated with connections inside an integrated circuit are negligible. So, you want to divide the network of gates such that the cost function C, the number of connections cutting across the partition, is minimized (see Figure 1). Because a million computers will be running almost non-stop for 10 years, removing even one costly connection would be worthwhile. Fortunately, this (simplified) problem can be mapped onto the well-known graph-bipartitioning problem. In this problem, the n gates are the vertices of a graph with edges between two connected gates. Each vertex is a Boolean variable, with state " 0 " if placed on the left integrated circuit and state " 1 " if placed on the right integrated circuit. Although the graph of connections is fixed, the vertices can be moved so that we may obtain a good partition. Unfortunately , optimizing the equal partition is NP-hard; that is, the computations needed to find the global optimum with certainty for even the cleverest algorithm grow faster than any power of n. This computation would become unreasonable for about n տ 10 3. Instead, we can " search " the space of all feasible (equal) partitions Ω. Because the configurations S ∈ Ω so far are unrelated , we need to define a " neighborhood " N(S) ⊂ Ω for each S, a way to proceed from the current configuration S to some neighboring configuration S′ ∈ N(S). 1 A simple neighborhood N for this problem is a " 1-exchange, " which consists of all S′ ∈ Ω obtained from S by changing a 0-vertex to 1 and a 1-vertex to 0 (to maintain an equal partition). The neighborhood N provides Ω with a metric such that the cost function C(S) exhibits local extrema, like a (high-dimensional) mountain landscape. Then, moving sequentially " downhill " to better configurations, we should reach a local minimum very quickly. However, in NP-hard optimization problems, the number of suboptimal minima of the cost function grows nearly as fast as the number of configurations , |Ω|, which here grows like. Thus, in this approach there is no way …