A Tight Runtime Analysis for the cGA on Jump Functions---EDAs Can Cross Fitness Valleys at No Extra Cost

We prove that the compact genetic algorithm (cGA) with hypothetical population size $\mu = \Omega(\sqrt n \log n) \cap \text{poly}(n)$ with high probability finds the optimum of any $n$-dimensional jump function with jump size $k < \frac 1 {20} \ln n$ in $O(\mu \sqrt n)$ iterations. Since it is known that the cGA with high probability needs at least $\Omega(\mu \sqrt n + n \log n)$ iterations to optimize the unimodal OneMax function, our result shows that the cGA in contrast to most classic evolutionary algorithms here is able to cross moderate-sized valleys of low fitness at no extra cost. Our runtime guarantee improves over the recent upper bound $O(\mu n^{1.5} \log n)$ valid for $\mu = \Omega(n^{3.5+\varepsilon})$ of Hasen\"ohrl and Sutton (GECCO 2018). For the best choice of the hypothetical population size, this result gives a runtime guarantee of $O(n^{5+\varepsilon})$, whereas ours gives $O(n \log n)$. We also provide a simple general method based on parallel runs that, under mild conditions, (i)~overcomes the need to specify a suitable population size, but gives a performance close to the one stemming from the best-possible population size, and (ii)~transforms EDAs with high-probability performance guarantees into EDAs with similar bounds on the expected runtime.Comment: 25 pages, full version of a paper to appear at GECCO 201

A Parameterized Complexity Analysis of Bi-level Optimisation with Evolutionary Algorithms

Bi-level optimisation problems have gained increasing interest in the field of combinatorial optimisation in recent years. With this paper, we start the runtime analysis of evolutionary algorithms for bi-level optimisation problems. We examine two NP-hard problems, the generalised minimum spanning tree problem (GMST), and the generalised travelling salesman problem (GTSP) in the context of parameterised complexity. For the generalised minimum spanning tree problem, we analyse the two approaches presented by Hu and Raidl (2012) with respect to the number of clusters that distinguish each other by the chosen representation of possible solutions. Our results show that a (1+1) EA working with the spanning nodes representation is not a fixed-parameter evolutionary algorithm for the problem, whereas the global structure representation enables to solve the problem in fixed-parameter time. We present hard instances for each approach and show that the two approaches are highly complementary by proving that they solve each other's hard instances very efficiently. For the generalised travelling salesman problem, we analyse the problem with respect to the number of clusters in the problem instance. Our results show that a (1+1) EA working with the global structure representation is a fixed-parameter evolutionary algorithm for the problem

Runtime analysis of evolutionary algorithms with complex fitness evaluation mechanisms

Evolutionary algorithms (EAs) are bio-inspired general purpose optimisation methods which are applicable to a wide range of problems. The performance of an EA can vary considerably according to the problem it tackles. Runtime analyses of EAs rigorously prove bounds on the expected computational resources required by the EA to solve a given problem. A crucial component of an EA is the way it evaluates the quality (i.e. fitness) of candidate solutions. Different fitness evaluation methods may drastically change the efficiency of a given EA. In this thesis, the effects of different fitness evaluation methods on the performance of evolutionary algorithms are investigated. A major contribution of this thesis is the first runtime analyses of EAs on bi-level optimisation problems. The performances of different EAs on The Generalised Minimum Spanning Tree Problem and The Generalised Travelling Salesperson Problem are analysed to illustrate how bi-level problem structures can be exploited to delegate part of the optimisation effort to problem-specific deterministic algorithms. Different bi-level representations are considered and it is proved that one of them leads to fixed-parameter evolutionary algorithms for both problems with respect to the number of clusters. Secondly, a new mathematical tool called the level-based theorem is presented. The theorem is a high level analytical tool which provides upper bounds on the runtime of a wide range of non-elitist population-based algorithms with independent sampling and using sophisticated high arity variation operators such as crossover. The independence of this new tool from the objective function allows runtime analyses of EAs which use complicated fitness evaluation methods. As an application of the level-based theorem, we conduct, for the first time, runtime analyses of non-elitist genetic algorithms on pseudo-Boolean test functions and also on three classical combinatorial optimisation problems. The last major contribution of this thesis is the illustration of how the level-based theorem can be used to design genetic algorithms with guaranteed runtime bounds. The well-known graph problems Single Source Shortest Path and All-Pairs Shortest Path are used as test beds. The used fitness evaluation method is tailored to incorporate the optimisation approach of a well known problem-specific algorithm and it is rigorously proved that the presented EA optimises both problems efficiently. The thesis is concluded with a discussion of the wider implications of the presented work and future work directions are explored

On Easiest Functions for Mutation Operators in Bio-Inspired Optimisation

Understanding which function classes are easy and which are hard for a given algorithm is a fundamental question for the analysis and design of bio-inspired search heuristics. A natural starting point is to consider the easiest and hardest functions for an algorithm. For the (1+1) EA using standard bit mutation (SBM) it is well known that OneMax is an easiest function with unique optimum while Trap is a hardest. In this paper we extend the analysis of easiest function classes to the contiguous somatic hypermutation (CHM) operator used in artificial immune systems. We define a function MinBlocks and prove that it is an easiest function for the (1+1) EA using CHM, presenting both a runtime and a fixed budget analysis. Since MinBlocks is, up to a factor of 2, a hardest function for standard bit mutations, we consider the effects of combining both operators into a hybrid algorithm. We rigorously prove that by combining the advantages of k operators, several hybrid algorithmic schemes have optimal asymptotic performance on the easiest functions for each individual operator. In particular, the hybrid algorithms using CHM and SBM have optimal asymptotic performance on both OneMax and MinBlocks. We then investigate easiest functions for hybrid schemes and show that an easiest function for an hybrid algorithm is not just a trivial weighted combination of the respective easiest functions for each operator.publishersversionPeer reviewe

A Parameterised Complexity Analysis of Bi-level Optimisation with Evolutionary Algorithms

Bi-level optimisation problems have gained increasing interest in the field of combinatorial optimisation in recent years. In this paper, we analyse the runtime of some evolutionary algorithms for bi-level optimisation problems. We examine two NP-hard problems, the generalised minimum spanning tree problem and the generalised travelling salesperson problem in the context of parameterised complexity. For the generalised minimum spanning tree problem, we analyse the two approaches presented by Hu and Raidl (2012) with respect to the number of clusters that distinguish each other by the chosen representation of possible solutions. Our results show that a (1+1) evolutionary algorithm working with the spanning nodes representation is not a fixed-parameter evolutionary algorithm for the problem, whereas the problem can be solved in fixed-parameter time with the global structure representation. We present hard instances for each approach and show that the two approaches are highly complementary by proving that they solve each other’s hard instances very efficiently. For the generalised travelling salesperson problem, we analyse the problem with respect to the number of clusters in the problem instance. Our results show that a (1+1) evolutionary algorithm working with the global structure representation is a fixed-parameter evolutionary algorithm for the problem

Runtime analysis of evolutionary algorithms with complex fitness evaluation mechanisms

Evolutionary algorithms (EAs) are bio-inspired general purpose optimisation methods which are applicable to a wide range of problems. The performance of an EA can vary considerably according to the problem it tackles. Runtime analyses of EAs rigorously prove bounds on the expected computational resources required by the EA to solve a given problem. A crucial component of an EA is the way it evaluates the quality (i.e. fitness) of candidate solutions. Different fitness evaluation methods may drastically change the efficiency of a given EA. In this thesis, the effects of different fitness evaluation methods on the performance of evolutionary algorithms are investigated. A major contribution of this thesis is the first runtime analyses of EAs on bi-level optimisation problems. The performances of different EAs on The Generalised Minimum Spanning Tree Problem and The Generalised Travelling Salesperson Problem are analysed to illustrate how bi-level problem structures can be exploited to delegate part of the optimisation effort to problem-specific deterministic algorithms. Different bi-level representations are considered and it is proved that one of them leads to fixed-parameter evolutionary algorithms for both problems with respect to the number of clusters. Secondly, a new mathematical tool called the level-based theorem is presented. The theorem is a high level analytical tool which provides upper bounds on the runtime of a wide range of non-elitist population-based algorithms with independent sampling and using sophisticated high arity variation operators such as crossover. The independence of this new tool from the objective function allows runtime analyses of EAs which use complicated fitness evaluation methods. As an application of the level-based theorem, we conduct, for the first time, runtime analyses of non-elitist genetic algorithms on pseudo-Boolean test functions and also on three classical combinatorial optimisation problems. The last major contribution of this thesis is the illustration of how the level-based theorem can be used to design genetic algorithms with guaranteed runtime bounds. The well-known graph problems Single Source Shortest Path and All-Pairs Shortest Path are used as test beds. The used fitness evaluation method is tailored to incorporate the optimisation approach of a well known problem-specific algorithm and it is rigorously proved that the presented EA optimises both problems efficiently. The thesis is concluded with a discussion of the wider implications of the presented work and future work directions are explored