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- W4297955215 abstract "In 2009, Roeglin and Teng showed that the smoothed number of Pareto optimal solutions of linear multi-criteria optimization problems is polynomially bounded in the number $n$ of variables and the maximum density $phi$ of the semi-random input model for any fixed number of objective functions. Their bound is, however, not very practical because the exponents grow exponentially in the number $d+1$ of objective functions. In a recent breakthrough, Moitra and O'Donnell improved this bound significantly to $O(n^{2d} phi^{d(d+1)/2})$. An intriguing problem, which Moitra and O'Donnell formulate in their paper, is how much further this bound can be improved. The previous lower bounds do not exclude the possibility of a polynomial upper bound whose degree does not depend on $d$. In this paper we resolve this question by constructing a class of instances with $Omega ((n phi)^{(d-log{d}) cdot (1-Theta{1/phi})})$ Pareto optimal solutions in expectation. For the bi-criteria case we present a higher lower bound of $Omega (n^2 phi^{1 - Theta{1/phi}})$, which almost matches the known upper bound of $O(n^2 phi)$." @default.
- W4297955215 created "2022-10-01" @default.
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- W4297955215 date "2010-12-06" @default.
- W4297955215 modified "2023-09-27" @default.
- W4297955215 title "Lower Bounds for the Smoothed Number of Pareto optimal Solutions" @default.
- W4297955215 doi "https://doi.org/10.48550/arxiv.1012.1163" @default.
- W4297955215 hasPublicationYear "2010" @default.
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