## Kd ai 1nN

Where N denotes the population size, i is the number of the tree that is actually generated (i 1, , N) and a, with 0 < a < (n 1) 2, is a parameter that controls the strength of the heuristic bias. The heuristic recombination operator is a modified version of KruskalRST* crossover. Firstly, the operator transfers all edges E1 n E2 that exist in both parents T and T2 to the offspring. Then, the remaining edges are chosen randomly from E' (E1 UE2) (E1 nE2) using a tournament with replacement...

## Uniform Redundancy

The NetKey encoding is a synonymously redundant encoding. To ensure that GEAs perform independently of the structure of the optimal solution, NetKeys should be uniformly redundant, i.e. unbiased. This section examines the bias of the NetKey encoding. We measure, in analogy to Sect. 6.3.3, for randomly created NetKey genotypes xgnd, the minimum phenotypic distance min(dprnd star) towards stars, and the average phenotypic distance dp,nd MST towards the MST. This is followed by empirical evidence...

## Genotypes and Phenotypes

In 1866, Mendel recognized that nature stores the complete genetic information for an individual in pairwise alleles (Mendel 1866). The genetic information that determines the properties, appearance, and shape of an individual is stored by a number of strings. Later, it was discovered that the genetic information is formed by a double string of four nucleotides, called DNA. Mendel realized that nature distinguishes between the genetic code of an individual and its outward appearance. The...

## Measurements of Problem Difficulty

In the previous section, we discussed the reasons for problem difficulty. In the following paragraphs, we describe some measurements of problem difficulty. To investigate how different representations influence the performance of GEAs, a measurement of problem difficulty is necessary. Based on the different reasons for problem difficulty which exist for different types of optimization methods, we discuss some common measurements of problem difficulty These four measurements of problem...

## Building Block Hypothesis

Using the definition of building blocks as being highly fit solutions to subproblems, the building block hypothesis can be formulated. It describes the processing of building blocks and is based on the quasi-decomposability of a problem (Goldberg 1989c, page 41) Short, low order, and highly fit schemata are sampled, recombined, and resampled to form strings of potentially higher fitness. The building block hypothesis basically states that GEAs mainly work due to their ability to propagate...

## Optimization Methods for the Optimal Communication Spanning Tree Problem

Like other constrained spanning tree problems, the OCST problem is NP-hard (Garey and Johnson 1979, p. 207). Furthermore, it was shown in Reshef (1999) that the problem is MAX SNP-hard (Papadimitriou and Yannakakis 1991) which means it cannot be solved using a polynomial-time approximation scheme, unless P NP. Therefore, the OCST problem belongs to the class of optimization problems that behave like MAX-3SAT (Garey and Johnson 1979). Only for a few easy and restricted problem instances have...

## Analysis of Representations

We briefly summarize the most important properties of the different tree representations and operators from the previous chapters. We focus on the results concerning redundancy, bias, scaling, and locality for Pr fer numbers, characteristic vectors (CV), the LNB encoding, NetKeys, NetDir, and edge-sets. For the use of Prufer numbers, NetKeys, CV, and NetDir no additional representation-specific parameters are necessary. In contrast, for LNBs and edge-sets, additional representation-specific...

## Ssct for t

For t < tdrift we assume no genetic drift, and for t > tdrift all the remaining bits are randomly fixed. Therefore, the probability of GA failure can be calculated as By using (3.19) and (3.20) the average percentage of incorrect alleles is adrift(A) 1 - - 14 For large N, no genetic drift occurs and we get the same failure probability as for the non-drift case (3.16). For small N the most salient I 2'fN I bits are solved correctly with probability 1 a and the...

## Binary String Representations

After we have defined the optimization problems, we present possible binary representations fg for integers. The representation fg assigns binary genotypes xg to integer phenotypes xp. Instead of using binary strings with cardinality x 2 for the genotypes, higher x-ary alphabets could also be used. Then, a x-ary alphabet is used for the string of length l instead of a binary alphabet. Therefore, instead of encoding 2l different individuals with a binary alphabet, we are able to encode X1...

## Experiments and Empirical Results

Here we present empirical results when using the binary trivial voting mapping for the one-max problem and the concatenated deceptive trap problem. The first test example for our empirical investigation is the one-max problem. This problem is very easy to solve for GEAs as the fitness of an individual is simply the number of ones in the binary phenotype. To ensure that recombination results in a proper mixing of the BBs, we use uniform crossover for all experiments with the one-max problem....

## Scalable Test Problems for Graphs

To examine the performance of optimization algorithms for the topological design of trees, standard test problems should be used. Motivated by the previous subsection, we define a fully easy and a fully difficult scalable tree problem. The one-max tree problem is based on the integer one-max problem (compare Sect. 5.1) (Ackley 1987). An optimal solution Topt is chosen either randomly or by hand. The structure of this optimal solution Topt can be determined It can be a star, a list, or a random...

## Constructing Trees from Random Keys

After we have presented RKs as the basis for the NetKey encoding, we still have to define a construction algorithm which creates a valid tree from a RK sequence. Both elements, the RKs and the construction algorithm are necessary for the new NetKey encoding. To get a synonymously and uniformly redundant encoding, we demand the construction algorithm to preserve the synonymity of the RK encoding and not to favor some phenotypes but to work uniformly. We have seen that we are able to give...

## Redundant and Unbiased Encoding

We illustrate that the CV encoding with repair mechanism is a redundant encoding. Furthermore, we show that when using the repair mechanism from the previous subsection the encoding is unbiased that means uniformly redundant. The genotype-phenotype mapping fg constructs a valid tree xp G < Pp with the help of the repair mechanism from every possible feasible or infeasible xg G g. This means that nn-2 phenotypes are represented by 2n(n-1) 2 genotypes. Therefore, the CV encoding is a redundant...

## Experimental Results

For our experiments we concatenate four instances of a deceptive trap problem for trees of size 3 and 4. The size of a sub-problem denotes the number of nodes. Therefore, the problems for the empirical investigation have either n x m 12 (size 3) or m x n 16 nodes (size 4). The minimum fitness of an individual is 0 and the maximum fitness is either 4 x 2 8 (4 instances of the 3 node trap) or 4 x 3 12 (four instances of the 4 node trap). In our experiments we use a simple genetic algorithm...

## Performance for the OCST Problem

We want to investigate the performance of GEAs using the different variants of the edge-set encoding for the OCST problem from Sect. 8.2.1. The OCST problem is defined as follows Let G (V, E) be a complete undirected graph with n V nodes and E edges. To every pair of nodes (ij) a non-negative distance weight dij and a non-negative communication requirement is associated. The communication cost c(T) (compare (8.2)) of a spanning tree T is defined as uristic muta mutation ( mutation ( mutation (...

## Empirical Results

In this subsection, we present an empirical investigation into how the problem complexity is changed for the one-max problem and the deceptive trap problem if low-locality representations are used. We experimentally show that for high-locality representations the fully easy one-max problem remains fully easy. In contrast, most of the low-locality representations make the one-max problem more difficult to solve for GAs. The situation is vice versa for the fully difficult deceptive trap where...

## Number of Neighbors

We have seen that Pr fer numbers only have high locality if they encode stars. Therefore, we focus on two issues Firstly, we want to reveal why Pr fer numbers have high locality when encoding stars. And secondly, we want to know how large the areas are of high locality. Finding answers for these questions helps us to more accurately predict the behavior of GEAs for different tree optimization problems. The investigation will show that the number of neighbors has a major impact on the answers to...