We study the approximability and the hardness of combinatorial multi-objective NP optimization problems (multi-objective problems, for short). Our contributions are: We define and compare several solution notions that capture reasonable algorithmic tasks for computing optimal solutions. These solution notions induce corresponding NP-hardness notions for which we prove implication and separation results. 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. © 2010 Springer-Verlag Berlin Heidelberg.
CITATION STYLE
Glaßer, C., Reitwießner, C., Schmitz, H., & Witek, M. (2010). Approximability and hardness in multi-objective optimization. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6158 LNCS, pp. 180–189). https://doi.org/10.1007/978-3-642-13962-8_20
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