Frahnow, Clemens; Kötzing, Timo **Ring Migration Topology Helps Bypassing Local Optima**Parallel Problem Solving From Nature (PPSN) 2018: 129–140

Running several evolutionary algorithms in parallel and occasionally exchanging good solutions is referred to as island models. The idea is that the independence of the different islands leads to diversity, thus possibly exploring the search space better. Many theoretical analyses so far have found a complete (or sufficiently quickly expanding) topology as underlying migration graph most efficient for optimization, even though a quick dissemination of individuals leads to a loss of diversity. We suggest a simple fitness function Fork with two local optima parametrized by \(r \geq 2\) and a scheme for composite fitness functions. We show that, while the (1+1) EA gets stuck in a bad local optimum and incurs a run time of \(\Theta(n^{2r})\) fitness evaluations on Fork, island models with a complete topology can achieve a run time of \(\Theta(n^{1.5r})\) by making use of rare migrations in order to explore the search space more effectively. Finally, the ring topology, making use of rare migrations and a large diameter, can achieve a run time of \(\tilde{\Theta}(n^r)\), the black box complexity of Fork. This shows that the ring topology can be preferable over the complete topology in order to maintain diversity.

Friedrich, Tobias; Göbel, Andreas; Quinzan, Francesco; Wagner, Markus **Heavy-tailed Mutation Operators in Single-Objective Combinatorial Optimization**Parallel Problem Solving From Nature (PPSN) 2018: 134–145

A core feature of evolutionary algorithms is their mutation operator. Recently, much attention has been devoted to the study of mutation operators with dynamic and non-uniform mutation rates. Following up on this line of work, we propose a new mutation operator and analyze its performance on the (1+1) Evolutionary Algorithm (EA). Our analyses show that this mutation operator competes with pre-existing ones, when used by the (1+1)EA on classes of problems for which results on the other mutation operators are available. We present a jump function for which the performance of the (1+1)EA using any static uniform mutation and any restart strategy can be worse than the performance of the (1+1)EA using our mutation operator with no restarts. We show that the (1+1)EA using our mutation operator finds a (1/3)-approximation ratio on any non-negative submodular function in polynomial time. This performance matches that of combinatorial local search algorithms specifically designed to solve this problem. Finally, we evaluate experimentally the performance of the (1+1)EA using our operator, on real-world graphs of different origins with up to \(\sim\)37,000 vertices and \(\sim\)1.6 million edges. In comparison with uniform mutation and a recently proposed dynamic scheme our operator comes out on top on these instances.

Kötzing, Timo; Krejca, Martin S. **First-Hitting Times Under Additive Drift**Parallel Problem Solving From Nature (PPSN) 2018: 92–104

For the last ten years, almost every theoretical result concerning the expected run time of a randomized search heuristic used drift theory, making it the arguably most important tool in this domain. Its success is due to its ease of use and its powerful result: drift theory allows the user to derive bounds on the expected first-hitting time of a random process by bounding expected local changes of the process – the drift. This is usually far easier than bounding the expected first-hitting time directly. Due to the widespread use of drift theory, it is of utmost importance to have the best drift theorems possible. We improve the fundamental additive, multiplicative, and variable drift theorems by stating them in a form as general as possible and providing examples of why the restrictions we keep are still necessary. Our additive drift theorem for upper bounds only requires the process to be nonnegative, that is, we remove unnecessary restrictions like a finite, discrete, or bounded search space. As corollaries, the same is true for our upper bounds in the case of variable and multiplicative drift

Kötzing, Timo; Krejca, Martin S. **First-Hitting Times for Finite State Spaces**Parallel Problem Solving From Nature (PPSN) 2018: 79–91

One of the most important aspects of a randomized algorithm is bounding its expected run time on various problems. Formally speaking, this means bounding the expected first-hitting time of a random process. The two arguably most popular tools to do so are the fitness level method and drift theory. The fitness level method considers arbitrary transition probabilities but only allows the process to move toward the goal. On the other hand, drift theory allows the process to move into any direction as long as it moves closer to the goal in expectation; however, this tendency has to be monotone and, thus, the transition probabilities cannot be arbitrary. We provide a result that combines the benefit of these two approaches: our result gives a lower and an upper bound for the expected first-hitting time of a random process over \(\{0, \ldots ,n\}\) that is allowed to move forward and backward by 1 and can use arbitrary transition probabilities. In case that the transition probabilities are known, our bounds coincide and yield the exact value of the expected first-hitting time. Further, we also state the stationary distribution as well as the mixing time of a special case of our scenario.

Kötzing, Timo; Lagodzinski, J. A. Gregor; Lengler, Johannes; Melnichenko, Anna **Destructiveness of Lexicographic Parsimony Pressure and Alleviation by a Concatenation Crossover in Genetic Programming**Parallel Problem Solving From Nature (PPSN) 2018: 42–54

For theoretical analyses there are two specifics distinguishing GP from many other areas of evolutionary computation. First, the variable size representations, in particular yielding a possible bloat (i.e. the growth of individuals with redundant parts). Second, the role and realization of crossover, which is particularly central in GP due to the tree-based representation. Whereas some theoretical work on GP has studied the effects of bloat, crossover had a surprisingly little share in this work. We analyze a simple crossover operator in combination with local search,where a preference for small solutions minimizes bloat (lexicographic parsimony pressure); the resulting algorithm is denoted ConcatenationCrossover GP. For this purpose three variants of the well-studied Majority test function with large plateaus are considered. We show that the Concatenation Crossover GP can efficiently optimize these test functions,while local search cannot be efficient for all three variants independent of employing bloat control.