a b s t r a c tQuadratic Knapsack Problem (QKP) extends the canonical simple Knapsack Problem where the value obtained by selecting a subset of objects is a function dependent not only on the value corresponding to individual objects selected but also on their pair-wise selection. QKP is NP Hard in stronger sense i.e. no pseudo-polynomial time algorithm is known to exist which can solve QKP instances. QKP has been studied intensively due to its simple structure yet challenging difficulty and numerous applications. Quantum Inspired Evolutionary Algorithm (QIEA) belongs to the class of Evolutionary Algorithms and exhibits behaviour of an Estimation of Distribution Algorithm (EDA). QIEA provides a generic framework that has to be carefully tailored for a given problem to obtain an effective implementation. Thus, several forms of QIEA exist in the literature. These have been successfully applied on many hard problems. A new QIEA, QIEA-PSA is proposed with improved exploration and exploitation capabilities. Computational experiments on these benchmarks show that QIEA-PSA is improved significantly both in terms of the quality of solutions and speed of convergence on several benchmark QKP instances. The ideas incorporated are general enough and can be utilized with advantage on other similar and not so similar problems.
This article describes how the 0/1 Multiple Knapsack Problem (MKP), a generalization of popular 0/1 Knapsack Problem, is NP-hard and harder than simple Knapsack Problem. Solution of MKP involves two levels of choice – one for selecting an item to be placed and the other for selecting the knapsack in which it is to be placed. Quantum Inspired Evolutionary Algorithms (QIEAs), a subclass of Evolutionary algorithms, have been shown to be effective in solving difficult problems particularly NP-hard combinatorial optimization problems. QIEAs provide a general framework which needs to be customized according to the requirements of a given problem to obtain good solutions in reasonable time. An existing QIEA for MKP (QIEA-MKP) is based on the representation where a Q-bit collapse into a binary number. But decimal numbers are required to identify the knapsack where an item is placed. The implementation based on such representation suffers from overhead of frequent conversion from binary numbers to decimal numbers and vice versa. The generalized QIEA (GQIEA) is based on a representation where a Q-bit can collapse into an integer and thus no inter conversion between binary and decimal is required. A set of carefully selected features have been incorporated in proposed GQIEA-MKP to obtain better solutions in lesser time. Comparison with QIEA-MKP shows that GQIEA-MKP outperforms it in providing better solutions in lesser time for large sized MKPs. The generalization proposed can be used with advantage in other Combinatorial Optimization problems with integer strings as solutions.
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