Analysis of the (1+1) EA on Subclasses of Linear Functions under Uniform and Linear Constraints

Abstract

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Runtime Analysis of Evolutionary Algorithms for the Knapsack Problem with Favorably Correlated Weights

“…With this paper, we have contributed to the theoretical understanding of evolutionary algorithms for constrained optimization problems by means of rigorous runtime analysis. We generalized the result for the (1+1) EA obtained for uniform weights given in [1] to favorably correlated weights. Furthermore, we investigated the multi-objective formulation of the knapsack problem.…”

“…It should be noted that Ω(n 2 ) is a lower bound for the (1+1) EA for knapsack instances with favorably correlated weights as this bounds already holds for special instances where the weights are all 1 [1]. The reason for this lower bound are special 2-bit flips that are necessary in the case that a current non-optimal solution has a maximal number of 1-bits.…”

“…We study the single-objective setting where the profit of the knapsack should be maximized subject to the weights of the items meeting a given capacity bound W. Furthermore, we investigate the multi-objective setting where the goal is to maximize the profit and minimize the weight of the chosen items at the same time. For the single-objective setting, we generalize the result on uniform weights given in [1] to knapsack instances with favorably correlated weights. Furthermore, we investigate the multi-objective problem of maximizing profit and minimizing weight for the knapsack problem with favorably correlated weights.…”

“…Recently, linear functions with a uniform constraint have been studied in [1]. This is equivalent to the knapsack problem where all items have a weight of 1.…”

“…The knapsack problem with favorably correlated weights is a generalization of the problem of optimizing a linear function with a uniform constraint investigated in [1] where we have w i = 1, 1 ≤ i ≤ n. As for the case of a uniform constraint, the knapsack problem with favorably correlated weights can be easily solved by a greedy algorithm which includes the items as they appear in the bit string. However, analyzing the behavior of evolutionary algorithms is interesting for this problem as it allows to understand the working behavior of this type of algorithms for the problem.…”