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A Dynamic Programming Heuristic for the Quadratic Knapsack Problem

Adam Letchford (a.n.letchford***at***lancaster.ac.uk)
Franklin Djeumou Fomeni (f.djeumoufomeni***at***lancaster.ac.uk)

Abstract: It is well known that the standard (linear) knapsack problem can be solved exactly by dynamic programming in O(nc) time, where n is the number of items and c is the capacity of the knapsack. The quadratic knapsack problem, on the other hand, is NP-hard in the strong sense, which makes it unlikely that it can be solved in pseudo-polynomial time. We show however that the dynamic programming approach to the linear knapsack problem can be modified to yield a highly effective constructive heuristic for the quadratic version. In our experiments, the lower bounds obtained by our heuristic were consistently within a fraction of a percent of optimal. Moreover, the addition of a simple local search step enabled us to obtain the optimal solution of all instances considered.

Keywords: knapsack problems, integer programming, dynamic programming

Category 1: Combinatorial Optimization

Category 2: Integer Programming ((Mixed) Integer Nonlinear Programming )

Category 3: Other Topics (Dynamic Programming )

Citation: Eventually published as: F. Djeumou Fomeni & A.N. Letchford (2014) A dynamic programming heuristic for the quadratic knapsack problem. INFORMS J. Comput., 26(1), 173–182.

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Entry Submitted: 03/17/2012
Entry Accepted: 03/17/2012
Entry Last Modified: 05/31/2014

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