Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Journal of the ACM (JACM)
Random knapsack in expected polynomial time
Journal of Computer and System Sciences - Special issue: STOC 2003
Multicriteria Optimization
The Smoothed Number of Pareto Optimal Solutions in Bicriteria Integer Optimization
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
Smoothed analysis: an attempt to explain the behavior of algorithms in practice
Communications of the ACM - A View of Parallel Computing
Smoothed Analysis of Multiobjective Optimization
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Pareto optimal solutions for smoothed analysts
Proceedings of the forty-third annual ACM symposium on Theory of computing
Improved smoothed analysis of multiobjective optimization
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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In 2009, Röglin 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 Ø 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(n2dØd(d+1)/2). An "intriguingproblem", which Moitra and O'Donnell formulate intheir 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 Ω((nØ)(d-log(d))ċ(1-Θ(1/Ø))) Pareto optimal solutions in expectation. For the bi-criteria case we present a higher lower bound of Ω(n2Ø1-Θ(1/Ø)), which almost matches the known upper bound of O(n2Ø).