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...