Solving the 0–1 Quadratic Knapsack Problem with a competitive Quantum Inspired Evolutionary Algorithm

Abstract

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.

Analysis and Acceleration of the Quadratic Knapsack Problem on an Ising Machine

“…The average success% over all instances for SA1 and SA2 is none against 6.7% for DA. Whereas in [8] success% of 100% for all instances within less than 0.02 seconds is achieved with QIEA.…”

“…where opt_val is the known optimal profit value in every QKP instance from [8] and obtainedValue is the value reached by DA or SA. "Instance name" is the name of the QKP instance used.…”

“…Various approaches ranging from exact algorithms, heuristics, meta-heuristics, linearization have been tried. The most efficient current approach to our knowledge comes from C. Patvardhan et al using a quantum inspired evolutionary algorithm (QIEA) [8] and will thus be the base for comparing our results. Although the QKP was first formulated as an Ising model in [9] by Glover et al, to our knowledge, results of running this Ising model on an Ising machine have yet to be published.…”

“…To our knowledge, since the publication of [2], quantum inspired evolution algorithm approach from Patvardhan et Al. [8] is the approach yielding the best results both time wise and chance of solving an instance, it will thus be used for comparison with our results.…”

“…This algorithm is a simplified version of "Procedure Improv-eLocal(P)" described in [8] adapted for an Ising machine running parallel trials on different replicas such as we have here. For all replicas' solution obtained, we try all possible item swaps from inside and outside the knapsack, do the swap which generates the largest profit increase (while not violating the weight constraint), and iterate until no improvement can be done, resulting in a computing intensive process to improve current solutions.…”