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Computers and Operations Research
Pareto Shortest Paths is Often Feasible in Practice
WAE '01 Proceedings of the 5th International Workshop on Algorithm Engineering
Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Journal of the ACM (JACM)
Random knapsack in expected polynomial time
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Typical Properties of Winners and Losers [0.2ex] in Discrete Optimization
SIAM Journal on Computing
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Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices
SIAM Journal on Matrix Analysis and Applications
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
Path trading: fast algorithms, smoothed analysis, and hardness results
SEA'11 Proceedings of the 10th international conference on Experimental algorithms
Lower bounds for the smoothed number of pareto optimal solutions
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
Finding short paths on polytopes by the shadow vertex algorithm
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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We present several new results about smoothed analysis of multiobjective optimization problems. Motivated by the discrepancy between worst-case analysis and practical experience, this line of research has gained a lot of attention in the last decade. We consider problems in which d linear and one arbitrary objective function are to be optimized over a set S⊆{0,1}n of feasible solutions. We improve the previously best known bound for the smoothed number of Pareto-optimal solutions to O(n2dφd), where φ denotes the perturbation parameter. Additionally, we show that for any constant c the c-th moment of the smoothed number of Pareto-optimal solutions is bounded by O((n2dφd)c). This improves the previously best known bounds significantly. Furthermore, we address the criticism that the perturbations in smoothed analysis destroy the zero-structure of problems by showing that the smoothed number of Pareto-optimal solutions remains polynomially bounded even for zero-preserving perturbations. This broadens the class of problems captured by smoothed analysis and it has consequences for non-linear objective functions. One corollary of our result is that the smoothed number of Pareto-optimal solutions is polynomially bounded for polynomial objective functions.