Abstract
We study the approximability and the hardness of combinatorial multi-objective NP optimization problems (multi-objective problems, for short). Our contributions are:
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We define and compare several solution notions that capture reasonable algorithmic tasks for computing optimal solutions.
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These solution notions induce corresponding NP-hardness notions for which we prove implication and separation results.
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We define approximative solution notions and investigate in which cases polynomial-time solvability translates from one to another notion. Moreover, for problems where all objectives have to be minimized, approximability results translate from single-objective to multi-objective optimization such that the relative error degrades only by a constant factor. Such translations are not possible for problems where all objectives have to be maximized (unless P = NP).
As a consequence we see that in contrast to single-objective problems (where the solution notions coincide), the situation is more subtle for multiple objectives. So it is important to exactly specify the NP-hardness notion when discussing the complexity of multi-objective problems.
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Glaßer, C., Reitwießner, C., Schmitz, H., Witek, M. (2010). Approximability and Hardness in Multi-objective Optimization. In: Ferreira, F., Löwe, B., Mayordomo, E., Mendes Gomes, L. (eds) Programs, Proofs, Processes. CiE 2010. Lecture Notes in Computer Science, vol 6158. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13962-8_20
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DOI: https://doi.org/10.1007/978-3-642-13962-8_20
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