A t-linearization to exactly solve 0-1 quadratic knapsack problems

This paper presents an exact solution method based on a new linearization scheme for the 0-1 Quadratic Knapsack Problem (QKP) which consists of maximizing a quadratic pseudo-Boolean function with non negative coefficients subject to a linear capacity constraint. Contrasting to traditional linearization schemes, our approach only adds one extra variable. We first convert (QKP) into an equivalent problem using only one additional real decision variable t and a quadratic constraint. We then replace the additional quadratic constraint by a set of linear constraints derived from the characterization of the induced integer quadric hull. The linear relaxation of the resulting problem (called the t-relaxation) provides an upper bound used in branch-and-bound scheme. This upper bound is numerically compared with the Billionnet et al. [2] bound, and the Branch-and-Bound scheme with the exact algorithm of Pisinger et al. [12]. The experiments show that our upper bound is competitive with the best upper bound method known [2] for (QKP) (less than 1% from the optimum). In addition, the proposed branch-andbound clearly outperforms the exact algorithm [12] for low density instances (25% and 50%) for all problem sizes up to 300 variables.