An effective GRASP and tabu search for the 0–1 quadratic knapsack problem

Abstract

As a generalization of the classical 0-1 knapsack problem, the 0-1 Quadratic Knapsack Problem (QKP) that maximizes a quadratic objective function subject to a linear capacity constraint is NP-hard in strong sense. In this paper, we propose a memory based Greedy Randomized Adaptive Search Procedures (GRASP) and a tabu search algorithm to find near optimal solution for the QKP. Computational experiments on benchmarks and on randomly generated instances demonstrate the effectiveness and the efficiency of the proposed algorithms, which outperforms the current state-of-the-art heuristic Mini-Swarm in computational time and in the probability to achieve optimal solutions. Numerical results on large-sized instances with up to 2000 binary variables have also been reported.

Introduction

The binary Quadratic Knapsack Problem (QKP) introduced by Gallo et al.[6] is a generalization of the classical 0-1 knapsack problem, which maximizes a quadratic objective function subject to a linear capacity constraint. The QKP can be formulated as

$$\{\begin{array}{l}\max {\sum }_{i=1}^n{\sum }_{j=1}^i{p}_{ij}{x}_i{x}_j\\ \mathrm{s}\mathrm{.t}.{\sum }_{j=1}^n{w}_j{x}_j\le c\\ x_j\in \{0,1\},j=1,\mathrm{..}.,n\end{array},$$

where $\{p_{ij}\}\in {\mathbf{{\rm Z}}}_{+}^{n\times n}$ is a symmetric profit matrix, $w_j\in {\mathbf{Z}}_{+}(j=1,\mathrm{..}.,n)$ is the weight of item j and $c\in {\mathbf{Z}}_{+}$ is the capacity of knapsack. $x_j\in \{0,1\}$$(j=1,\mathrm{..}.,n)$ is a binary decision variable, which is equal to 1 if item j is selected and 0 otherwise.

The QKP has inspired intensive studies due to its simple structure but NP-hardness in computational complexity theory and its wide applicability in graph theory problems (max clique [4], weighted maximum b-clique problem [15]), in facility location problems [19], [20], in complier design [11] and in separation of valid inequalities (rounded capacity inequalities, path inequalities and depot capacity inequalities) for vehicle routing problems [26].

Different techniques and approaches have been proposed for the QKP. To our best knowledge, most of the exact methods are designed under a branch-and-bound (B&B) framework, where numerous upper bounds have been derived by using upper planes [6], linearization [11], [1], reformulation [3], Lagrangian relaxation [3], Lagrangian decomposition [14], [2], and semidefinite programming [10]. It is worth mentioning that the exact algorithm developed by Pisinger et al. [17] based on Lagrangian relaxation\decomposition and aggressive reduction is one of the most effective exact algorithms for the QKP up-to-date, in which some large-sized instances with up to 1500 binary variables had been solved to optimality within a reasonable time. Recently, Létocart et al. [13] proposed a reoptimization technique to accelerate the resolution of each independent sequence of 0-1 linear knapsack problems induced by the Lagrangian relaxation\decomposition [2], [3]. Computational results on randomly generated instances with up to 600 binary variables were reported.

Heuristic methods have also been proposed for the sake of practical interest, which require finding good feasible solution in high probability within reasonable time. Hammer and Rader [9] proposed a linearization and exchange (LEX) heuristic, which firstly finds a good initial solution by an approximation method and then improves this solution through an exchange method. Julstorm [12] compared the performance of several heuristics, including greedy, genetic and greedy genetic algorithm for the QKP. A dynamic programming heuristic was recently proposed by Fomeni and Letchford [27]. The Mini-Swarm algorithm proposed by Xie and Liu [21] is the current state-of-the-art heuristic, which computationally proves to be highly effective and capable of solving to optimality a very high percentage (94.9%) of Billionnet and Soutif's benchmark instances [2] with up to 200 binary variables in a very reasonable time. Approximation algorithms had also been proposed for special cases of QKP. Rader and Woeginger [28] developed a FPTAS for the special case where all profits $p_{ij}\ge 0$ and where the underlying graph is so-called edge series parallel. Recently, Kellerer and Strusevich [29], and Xu [30] proposed a FPTAS for the symmetric quadratic knapsack problem (SQKP), where the objective function can be separated into two terms, one depending on the variables $x_j$, and the other depending on the variables $(1-x_j)$, $j=1,\mathrm{.}..,n$. For a systematic survey on resolution methodologies of QKP and its applications, we recommend readers to refer [16].

Note that those large-sized benchmark instances of Pisinger et al. [17] (1500 binary variables) and Létocart et al. [13] (600 binary variables) were randomly generated and tested. They had not been recorded by the authors. (We had contacted Prof. Pisinger and Porf. Létocart). To push ahead the study on the resolution of QKP, the objective of this paper is thus to propose an effective heuristic to near optimal solution (or lower bound) for the QKP, since high quality solution is very necessary in exact algorithm development, especially, for solving those large-sized instances.

The Greedy Randomized Adaptive Search Procedures (GRASP) proposed by Feo and Resende [5] is an iterative multi-start process for combinatorial optimization problems, which has the characteristic of easy implementation. The GRASP has been applied successfully in a wide range of problem areas [18], [22]. In this paper, we propose a new memory based GRASP for the QKP. Different from those memory-based mechanisms proposed in literatures, for example Fleurent and Glover [23], and Prais and Ribeiro [24], in the construction phase of the proposed GRASP, the algorithm tends to select those high quality elements that have relatively little been selected in local optima obtained in previous GRASP iterations. A short-term memory tabu search is also proposed to further improve the best solution found so far by GRASP. Under the observation that the local optima have some similarities, the GRASP is restarted with those items that have always been selected in previously obtained local optima selected while in constructing new feasible solution in subsequent construction phase of GRASP iterations. The restart of GRASP is terminated if the best solution found so far has not been improved for a given number of consecutive restarts. By doing so, we can on the one hand reduce computational efforts in building new initial solution for local search phase of GRASP, and on the other hand focus more attention on the solution space with those items selected. Numerical experiments on benchmark instances demonstrate the effectiveness and the efficiency of the proposed algorithm, which outperforms the state-of-art heuristic Mini-Swarm proposed by Xie and Liu [21] in computational time and in the probability to attain optimal solutions. Numerical results on randomly generated instances with up to 2000 binary variables have also been reported. Note in literatures of QKP, the most large-sized benchmark instances have 1500 binary variables [17].

The remainder of the paper is organized as follows: the fundamental principles of GRASP are introduced in Section 2. The proposed memory based GRASP with restart is discussed in Section 3. A tabu search procedure is described in Section 4. Computational results on benchmark instances and on randomly generated instances are reported in Section 5. Finally, a conclusion is given in Section 6.