Abstract
We rigorously analyze the runtime of evolutionary algorithms for the classical knapsack problem where the weights are favorably correlated with the profits. Our result for the (\(1+1\)) EA generalizes the one obtained in [1] for uniform constraints and shows that an optimal solution in the single-objective setting is obtained in expected time \(O(n^2( \log n + \log p_{\max }))\), where \(p_{\max }\) is the largest profit of the given input. Considering the multi-objective formulation where the goal is to maximize the profit and minimize the weight of the chosen item set at the same time, we show that the Pareto front has size \(n+1\) whereas there are sets of solutions of exponential size where all solutions are incomparable to each other. Analyzing a variant of GSEMO with a size-based parent selection mechanism motivated by these insights, we show that the whole Pareto front is computed in expected time \(O(n^3)\).
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This work has been supported through Australian Research Council (ARC) grant DP160102401.
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Neumann, F., Sutton, A.M. (2018). Runtime Analysis of Evolutionary Algorithms for the Knapsack Problem with Favorably Correlated Weights. In: Auger, A., Fonseca, C., Lourenço, N., Machado, P., Paquete, L., Whitley, D. (eds) Parallel Problem Solving from Nature – PPSN XV. PPSN 2018. Lecture Notes in Computer Science(), vol 11102. Springer, Cham. https://doi.org/10.1007/978-3-319-99259-4_12
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