A new upper bound for the 0-1 quadratic knapsack problem

Billionnet, Alain; Faye, Alain; Soutif, Éric

doi:10.1016/s0377-2217(97)00414-1

Cited by 39 publications

(29 citation statements)

References 11 publications

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Section: Introductionmentioning

confidence: 99%

“…Classical exact methods [39] for solving the QKP are the branch and bound (B&B) algorithms, where numerous upper bounds have been obtained, by using techniques such as derivation of upper planes [20], Lagrangian relaxation [24], reformulation [10], linearization [3], Lagrangian decomposition [4,5,41], semidefinite relaxation [26], and reduction strategies [24,45], etc.…”

Section: Introductionmentioning

confidence: 99%

“…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].…”

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confidence: 99%

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A Mini-Swarm for the Quadratic Knapsack Problem

Xie

Liu

2007

2007 IEEE Swarm Intelligence Symposium

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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...

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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A Mini-Swarm for the Quadratic Knapsack Problem

Xie

Liu

2007

2007 IEEE Swarm Intelligence Symposium

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“…It may be stated mathematically as follows: The problem (QMKP) which is a NP-hard problem [3] is a generalization of both the integer quadratic knapsack problem [2] and the 0-1 quadratic knapsack problem where the objective function is subject to only one constraint [1].…”

Section: Introductionmentioning

confidence: 99%

Rewriting integer variables into zero-one variables: Some guidelines for the integer quadratic multi-knapsack problem

Quadri

Soutif

2007

Oper Res Int J

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This paper is concerned with the integer quadratic multidimensional knapsack problem (QMKP) where the objective function is separable. Our objective is to determine which expansion technique of the integer variables is the most appropriate to solve (QMKP) to optimality using the upper bound method proposed by Quadri et al. (2007). To the best of our knowledge the upper bound method previously mentioned is the most effective method in the literature concerning (QMKP). This bound is computed by transforming the initial quadratic problem into a 0-1 equivalent piecewise linear formulation and then by establishing the surrogate problem associated. The linearization method consists in using a direct expansion initially suggested by Glover (1975) of the integer variables and in applying a piecewise interpolation to the separable objective function. As the direct expansion results in an increase of the size of the problem, other expansions techniques may be utilized to reduce the number of 0-1 variables so as to make easier the solution to the linearized problem. We will compare theoretically the use in the upper bound process of the direct expansion (I) employed in Quadri et al. (2007) with two other basic expansions, namely: (II) a direct expansion with additional constraints and (III) a binary expansion. We show that expansion (II) provides a bound which value is equal to the one computed by Quadri et al (2007). Conversely, we provide the proof of the non applicability of expansion (III) in the upper bound method. More specifically, we will show that if (III) is used to rewrite the integer variables into 0-1 variables then a 2 linear interpolation can not be applied to transform (QMKP) into an equivalent 0-1 piecewise linear problem.

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“…Exact algorithms for this problem include Michelon and Veuilleux [13], Hammer and Rader [14], Billionnet et al [15], Pisinger et al [16] and Caprara et al [17]. The latter algorithm is based on branch-and-bound search where tight bounds are found through a reformulation.…”

Section: Introductionmentioning

confidence: 99%

Upper bounds and exact algorithms for -dispersion problems

Pisinger

2006

Computers & Operations Research

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A new upper bound for the 0-1 quadratic knapsack problem

Cited by 39 publications

References 11 publications

A Mini-Swarm for the Quadratic Knapsack Problem

A Mini-Swarm for the Quadratic Knapsack Problem

Rewriting integer variables into zero-one variables: Some guidelines for the integer quadratic multi-knapsack problem

Upper bounds and exact algorithms for -dispersion problems

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