A Mini-Swarm for the Quadratic Knapsack Problem

Abstract

Abstract-The 0-1 quadratic knapsack problem (QKP) is a hard computational problem, which is a generalization of the knapsack problem (KP). In this paper, a mini-Swarm system is presented. Each agent, realized with minor declarative knowledge and simple behavioral rules, searches on a structural landscape of the problem through the guided generate-and-test behavior under the law of socially biased individual learning, and cooperates with others by indirect interactions. The formal decomposition of behaviors allows understanding and reusing elemental operators, while utilizes the heuristic information on the landscape. The results on a collection of the QKP instances by mini-Swarm versions are compared with that of both a branch-and-bound algorithm and a greedy genetic algorithm, which show its effectiveness. I. INTRO DUCTIONThe 0-1 quadratic knapsack problem (QKP) was introduced by Gallo et al. [20], which consists in choosing elements from n items for maximizing a quadratic profit objective function subject to a linear capacity constraint. The QKP is a generalization of the 0-1 knapsack problem (KP) [38], by restricting all the quadratic coefficients to zero. The knapsack problems [38] have been intensively studied due to both its theoretical interest and its wide practical applicability. Due to its generality, the QKP has more practically applications [3,20], moreover, several graph problems can be formulated as the QKP [4,45].The KP is NP-hard, although it can be solved exactly in pseudo-polynomial time through dynamic programming based on Bellman recursion [21]. As a generalized version, the QKP is much harder than the KP [10,45], where the graph structure associated with coefficients of objective function plays a important role in its solution [46].Due to the NP-hardness in strong sense, the research efforts on solving the QKP may focus on exact methods, approximate algorithms and heuristic methods.Classical The studies on approximate algorithms, which seek guaranteed polynomial-time performance on approximate solution, were only limited on some special cases of the QKP, such as the classical KP [30], and the QKP where the underlying graph is edge series-parallel [46], etc.The heuristic methods arise from practical interest, which try to find good solution in high probability with acceptable computational efforts by considering certain imprecise knowledge on the structure of the problem. The linearization and exchange (LEX) heuristic [24] uses the best linear L2-approximation to build an associated linear KP and use it for determining items in a greedy way to form a good initial solution, which is then improved by an exchange method between certain items. Hua et al. [29] studied on the convex QKP by an approximate dynamic programming approach. Julstrom [32] also illustrated the capability of several heuristics, including greedy, genetic and greedy genetic algorithms on the QKP.Swarm algorithms [7,16,34] address on tackling a specified optimization task by multiple interacting agents, where each agent works u...

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

“…The ABC [15] algorithm has been shown to outperform Mini-Swarm [14] both in terms of probability to reach the optimal and time taken to compute them. Mini-Swarm provides 7590 hits to optimal out of 8000 trials for 100 and 200 object instances while ABC hits optimal 7892 times.…”

“…Julstrom [12] presented studies on various greedy and genetic algorithms and has shown a greedy genetic algorithm (GGA) to solve a sample of Billionet and Soutif's benchmark instances [13] (BS benchmark instances) with up to 200 variables to optimality in 902 out of 1000 trials using around 15000 Function Evaluations (FES) on an average per trial. A Mini-Swarm algorithm is proposed by Xie and Liu [14] and an artificial bee colony (ABC) algorithm by Pulikanti and Singh [15] which are shown to solve all BS benchmark instances with up to 200 binary variables to optimality with high probability (0.949 and 0.987 respectively) in a very reasonable time. The ABC algorithm has been shown to outperform Mini-Swarm in [15].…”

“…The computational effort is reported as time in seconds for heuristics while for GGA both the generations required to reach optimal and time in seconds to reach optimal are reported. Xie and Liu [14] reported number of hits made to optimal in 100 runs and average execution time per run in seconds. Pulikanti and Singh [15] reported number of hits made to optimal and average value obtained in 100 runs for each problem for quality of solution and minimum and average time to reach optimal.…”

Towards the right amount of randomness in quantum-inspired evolutionary algorithms

“…A Mini-Swarm algorithm is proposed by Xie and Liu (2007) is shown to solve all BS benchmark instances with 100 and 200 binary variables to optimality with high probability in a reasonable time. Patvardhan et al (2012) presented known best QIEA for QKP (dubbed QIEA-PPA in this paper).…”

“…It is clear that QIEA-QKP outperforms both QIEA-PPA and GGA in terms of frequency of reaching optimal and computational effort required. Table 9 shows the comparison of proposed QIEA-QKP with the a popular population-based Mini-Swarm algorithm given by Xie and Liu (2007) using BS benchmark instances of size 100 and 200 variables. Table 9 presents number of times optimal solution is reached in 100 runs (Hits), average value over 100 runs of relative percentage deviation (RPD) from the optimal and average time taken (AvgT) in seconds required to reach best solution.…”