A Study on the Computational Complexity of the Bilevel Knapsack Problem

We analyze the computational complexity of three fundamental variants of the bilevel knapsack problem. All three variants are shown to be complete for the second level of the polynomial hierarchy. We also discuss the somewhat easier situation where the weight and profit coefficients in the knapsack...

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Published in:SIAM journal on optimization Vol. 24; no. 2; pp. 823 - 838
Main Authors: Caprara, Alberto, Carvalho, Margarida, Lodi, Andrea, Woeginger, Gerhard J.
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2014
Subjects:
Algorithms
Applied mathematics
Approximation
Complexity
Complexity theory
Computation
Decision making
Hierarchies
Knapsack problem
Mathematical analysis
Optimization
Polynomials
ISSN:1052-6234, 1095-7189
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Abstract We analyze the computational complexity of three fundamental variants of the bilevel knapsack problem. All three variants are shown to be complete for the second level of the polynomial hierarchy. We also discuss the somewhat easier situation where the weight and profit coefficients in the knapsack problem are encoded in unary: two of the considered bilevel variants become solvable in polynomial time, whereas the third becomes 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\;$\ne$\;NP). [PUBLICATION ABSTRACT]
AbstractList We analyze the computational complexity of three fundamental variants of the bilevel knapsack problem. All three variants are shown to be complete for the second level of the polynomial hierarchy. We also discuss the somewhat easier situation where the weight and profit coefficients in the knapsack problem are encoded in unary: two of the considered bilevel variants become solvable in polynomial time, whereas the third becomes 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\; $\ne$\; NP).
We analyze the computational complexity of three fundamental variants of the bilevel knapsack problem. All three variants are shown to be complete for the second level of the polynomial hierarchy. We also discuss the somewhat easier situation where the weight and profit coefficients in the knapsack problem are encoded in unary: two of the considered bilevel variants become solvable in polynomial time, whereas the third becomes 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\;$\ne$\;NP). [PUBLICATION ABSTRACT]
Author Lodi, Andrea
Caprara, Alberto
Carvalho, Margarida
Woeginger, Gerhard J.
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  givenname: Gerhard J.
  surname: Woeginger
  fullname: Woeginger, Gerhard J.
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10.1016/j.tcs.2012.08.008
10.1016/0304-3975(76)90061-X
10.1007/BF01096458
10.1112/jlms/s1-16.4.212
10.1007/BF01586088
10.1016/j.orl.2009.01.007
10.1080/0233193031000149894
10.1016/j.tcs.2007.03.006
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SubjectTerms Algorithms
Applied mathematics
Approximation
Complexity
Complexity theory
Computation
Decision making
Hierarchies
Knapsack problem
Mathematical analysis
Optimization
Polynomials
Title A Study on the Computational Complexity of the Bilevel Knapsack Problem
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