# Algorithmic improvements on dynamic programming for the bi-objective {0,1} knapsack problem

## Abstract

This paper presents several methodological and algorithmic improvements over a state-of-the-art dynamic programming algorithm for solving the bi-objective {0,1} knapsack problem. The variants proposed make use of new definitions of lower and upper bounds, which allow a large number of states to be discarded. The computation of these bounds are based on the application of dichotomic search, definition of new bound sets, and bi-objective simplex algorithms to solve the relaxed problem. Although these new techniques are not of a common application for dynamic programming, we show that the best variants tested in this work can lead to an average improvement of 10 to 30 % in CPU-time and significant less memory usage than the original approach in a wide benchmark set of instances, even for the most difficult ones in the literature.

## Keywords

Bi-objective 0-1 knapsack problems Multi-objective combinatorial optimization Bounds sets Dichotomic search Bi-objective simplex algorithm## Notes

### Acknowledgements

The authors acknowledge the anonymous referee that suggested the second version of variant B-DP3. J.R. Figueira, L. Paquete and D. Vanderpooten acknowledge the financial support from the Luso-French bilateral cooperation agreement between LAMSADE and CEG-IST (FCT/CNRS 2009). J.R. Figueira and M. Simões acknowledge a grant from CEG-IST, Instituto Superior Técnico (PTDC/GES/73853/2006). D. Vanderpooten acknowledges the support by the project ANR-09-BLAN-0361 “GUaranteed Efficiency for PAReto optimal solutions Determination (GUEPARD)”.

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