A complexity and approximability study of the bilevel knapsack problem

Authors:
Alberto Caprara;Margarida Carvalho;Andrea Lodi;Gerhard J. Woeginger
Affiliations:
DEI, University of Bologna, Italy;Departamento de Ciências de Computadores, Universidade do Porto, Portugal;DEI, University of Bologna, Italy;Department of Mathematics, TU Eindhoven, Netherlands
Venue:
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
Year:
2013

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Abstract

We analyze three fundamental variants of the bilevel knapsack problem, which all are complete for the second level of the polynomial hierarchy. If the weight and profit coefficients in the knapsack problem are encoded in unary, then two of the bilevel variants are solvable in polynomial time, whereas the third is NP-complete. Furthermore we design a polynomial time approximation scheme for this third variant, whereas the other two variants cannot be approximated in polynomial time within any constant factor (assuming P≠NP).