Exact Markov chain-based runtime analysis of a discrete particle swarm optimization algorithm on sorting and OneMax

Mühlenthaler, Moritz; Raß, Alexander; Schmitt, Manuel; Wanka, Rolf

doi:10.1007/s11047-021-09856-0

“…Finally, in [7] it is discussed how to adjust the (1 + ( , )) GA to permutation spaces and then an ( 2 ) runtime of the resulting algorithm on the sorting problem with H fitness is proven. Slightly less related to the focus of this work, there is an interesting a sequence of results on how EAs optimize NP-hard variants of the travelling salesman problem (TSP) in the parameterized complexity paradigm [10,45,46], works on finding diverse sets of TSP solutions [14,15], a fixed-budget analysis for the TSP [36], and a result on how particle swarm algorithms solve the sorting problem [35].…”

Section: Previous Workmentioning

confidence: 99%

Towards a stronger theory for permutation-based evolutionary algorithms

Doerr

¹

,

Ghannane

²

,

Brahim

³

2022

Proceedings of the Genetic and Evolutionary Computation Conference

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While the theoretical analysis of evolutionary algorithms (EAs) has made significant progress for pseudo-Boolean optimization problems in the last 25 years, only sporadic theoretical results exist on how EAs solve permutation-based problems.To overcome the lack of permutation-based benchmark problems, we propose a general way to transfer the classic pseudo-Boolean benchmarks into benchmarks defined on sets of permutations. We then conduct a rigorous runtime analysis of the permutation-based (1 + 1) EA proposed by Scharnow, Tinnefeld, and Wegener ( 2004) on the analogues of the L O and J benchmarks. The latter shows that, different from bit-strings, it is not only the Hamming distance that determines how difficult it is to mutate a permutation into another one , but also the precise cycle structure of −1 . For this reason, we also regard the more symmetric scramble mutation operator. We observe that it not only leads to simpler proofs, but also reduces the runtime on jump functions with odd jump size by a factor of Θ( ). Finally, we show that a heavy-tailed version of the scramble operator, as in the bit-string case, leads to a speed-up of order Θ( ) on jump functions with jump size . CCS CONCEPTS• Theory of computation → Theory of randomized search heuristics.

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“…Finally, in [7] it is discussed how to adjust the (1 + ( , )) GA to permutation spaces and then an ( 2 ) runtime of the resulting algorithm on the sorting problem with H fitness is proven. Slightly less related to the focus of this work, there is an interesting a sequence of results on how EAs optimize NP-hard variants of the travelling salesman problem (TSP) in the parameterized complexity paradigm [10,45,46], works on finding diverse sets of TSP solutions [14,15], a fixed-budget analysis for the TSP [36], and a result on how particle swarm algorithms solve the sorting problem [35].…”

Section: Previous Workmentioning

confidence: 99%

Towards a Stronger Theory for Permutation-based Evolutionary Algorithms

Doerr,

Ghannane,

Brahim

2022

Preprint

0

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While the theoretical analysis of evolutionary algorithms (EAs) has made significant progress for pseudo-Boolean optimization problems in the last 25 years, only sporadic theoretical results exist on how EAs solve permutation-based problems.To overcome the lack of permutation-based benchmark problems, we propose a general way to transfer the classic pseudo-Boolean benchmarks into benchmarks defined on sets of permutations. We then conduct a rigorous runtime analysis of the permutation-based (1 + 1) EA proposed by Scharnow, Tinnefeld, and Wegener ( 2004) on the analogues of the L O and J benchmarks. The latter shows that, different from bit-strings, it is not only the Hamming distance that determines how difficult it is to mutate a permutation into another one , but also the precise cycle structure of −1 . For this reason, we also regard the more symmetric scramble mutation operator. We observe that it not only leads to simpler proofs, but also reduces the runtime on jump functions with odd jump size by a factor of Θ( ). Finally, we show that a heavy-tailed version of the scramble operator, as in the bit-string case, leads to a speed-up of order Θ( ) on jump functions with jump size . CCS CONCEPTS• Theory of computation → Theory of randomized search heuristics.

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“…Performance analysis of metaheuristics are not simple, since it is highly random in nature. MCs are one of the widely used methods to analyze the performance of metaheuristics [17][18][19][20][21][22][23][24][25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Search Graph Magnification in Rapid Mixing of Markov Chains Associated with the Local Search-Based Metaheuristics

Shenoy

¹

,

Pai

²

2021

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The structural property of the search graph plays an important role in the success of local search-based metaheuristic algorithms. Magnification is one of the structural properties of the search graph. This study builds the relationship between the magnification of a search graph and the mixing time of Markov Chain (MC) induced by the local search-based metaheuristics on that search space. The result shows that the ergodic reversible Markov chain induced by the local search-based metaheuristics is inversely proportional to magnification. This result indicates that it is desirable to use a search space with large magnification for the optimization problem in hand rather than using any search spaces. The performance of local search-based metaheuristics may be good on such search spaces since the mixing time of the underlying Markov chain is inversely proportional to the magnification of search space. Using these relations, this work shows that MC induced by the Metropolis Algorithm (MA) mixes rapidly if the search graph has a large magnification. This indicates that for any combinatorial optimization problem, the Markov chains associated with the MA mix rapidly i.e., in polynomial time if the underlying search graph has large magnification. The usefulness of the obtained results is illustrated using the 0/1-Knapsack Problem, which is a well-studied combinatorial optimization problem in the literature and is NP-Complete. Using the theoretical results obtained, this work shows that Markov Chains (MCs) associated with the local search-based metaheuristics like random walk and MA for 0/1-Knapsack Problem mixes rapidly.

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Exact Markov chain-based runtime analysis of a discrete particle swarm optimization algorithm on sorting and OneMax

Cited by 6 publications

References 38 publications

Towards a stronger theory for permutation-based evolutionary algorithms

Towards a stronger theory for permutation-based evolutionary algorithms

Towards a Stronger Theory for Permutation-based Evolutionary Algorithms

Search Graph Magnification in Rapid Mixing of Markov Chains Associated with the Local Search-Based Metaheuristics

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