Abstract
This paper investigates, for the first time in the literature, the approximation of min-max (regret) versions of classical problems like shortest path, minimum spanning tree, and knapsack. For a bounded number of scenarios, we establish fully polynomial-time approximation schemes for the min-max versions of these problems, using relationships between multi-objective and min-max optimization. Using dynamic programming and classical trimming techniques, we construct a fully polynomial-time approximation scheme for min-max regret shortest path. We also establish a fully polynomial-time approximation scheme for min-max regret spanning tree and prove that min-max regret knapsack is not at all approximable. We also investigate the case of an unbounded number of scenarios, for which min-max and min-max regret versions of polynomial-time solvable problems usually become strongly NP-hard. In this setting, non-approximability results are provided for min-max (regret) versions of shortest path and spanning tree.
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Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and approximation. In: Combinatorial optimization problems and their approximability properties. Springer, Heidelberg (1999)
Garey, M., Johnson, D.: Computer and Intractability: A Guide to the theory of NP-completeness. Freeman, New York (1979)
Kouvelis, P., Yu, G.: Robust Discrete Optimization and its Applications. Kluwer Academic Publishers, Boston (1997)
Mahajan, M., Vinay, V.: Determinants: combinatorics, algorithms, and complexity. In: Proceedings of the Eigth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, New Orleans, USA, pp. 730–738 (1997)
Papadimitriou, C.H., Yannakakis, M.: On the approximability of trade-offs and optimal access of web sources. In: IEEE Symposium on Foundations of Computer Science, pp. 86–92 (2000)
Sahni, S.: General techniques for combinatorial approximation. Operations Research 25(6), 920–936 (1977)
Vazirani, V.V.: Approximation Algorithms. Springer, Heidelberg (2001)
Zuckerman, D.: NP-complete problems have a version that’s hard to approximate. In: Proceeding 8th Annual Conference on Structure in Complexity Theory, pp. 305–312 (1993)
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© 2005 Springer-Verlag Berlin Heidelberg
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Aissi, H., Bazgan, C., Vanderpooten, D. (2005). Approximation Complexity of min-max (Regret) Versions of Shortest Path, Spanning Tree, and Knapsack. In: Brodal, G.S., Leonardi, S. (eds) Algorithms – ESA 2005. ESA 2005. Lecture Notes in Computer Science, vol 3669. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11561071_76
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DOI: https://doi.org/10.1007/11561071_76
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-29118-3
Online ISBN: 978-3-540-31951-1
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