Matching is as easy as matrix inversion
Combinatorica
Exact arborescences, matchings and cycles
Discrete Applied Mathematics
Approximation of Pareto optima in multiple-objective, shortest-path problems
Operations Research
The complexity of restricted spanning tree problems
Journal of the ACM (JACM)
Improved bounds on planar k-sets and k-levels
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Parametric and Kinetic Minimum Spanning Trees
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
On the approximability of trade-offs and optimal access of Web sources
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Journal of the ACM (JACM)
Typical properties of winners and losers in discrete optimization
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Random knapsack in expected polynomial time
Journal of Computer and System Sciences - Special issue: STOC 2003
Multicriteria Optimization
Efficiently computing succinct trade-off curves
Theoretical Computer Science - Automata, languages and programming: Algorithms and complexity (ICALP-A 2004)
Stochastic shortest paths via Quasi-convex maximization
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
Small Approximate Pareto Sets for Bi-objective Shortest Paths and Other Problems
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Approximation algorithms for reliable stochastic combinatorial optimization
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Timetable information: models and algorithms
ATMOS'04 Proceedings of the 4th international Dagstuhl, ATMOS conference on Algorithmic approaches for transportation modeling, optimization, and systems
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
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We consider bicriteria optimization problems and investigate the relationship between two standard approaches to solving them: (i) computing the Pareto curve and (ii) the so-called decision maker's approach in which both criteria are combined into a single (usually non-linear) objective function. Previous work by Papadimitriou and Yannakakis showed how to efficiently approximate the Pareto curve for problems like Shortest Path, Spanning Tree, and Perfect Matching. We wish to determine for which classes of combined objective functions the approximate Pareto curve also yields an approximate solution to the decision maker's problem. We show that an FPTAS for the Pareto curve also gives an FPTAS for the decision maker's problem if the combined objective function is growth bounded like a quasi-polynomial function. If these functions, however, show exponential growth then the decision maker's problem is NP-hard to approximate within any factor. In order to bypass these limitations of approximate decision making, we turn our attention to Pareto curves in the probabilistic framework of smoothed analysis. We show that in a smoothed model, we can efficiently generate the (complete and exact) Pareto curve with a small failure probability if there exists an algorithm for generating the Pareto curve whose worst case running time is pseudopolynomial. This way, we can solve the decision maker's problem w.r.t. any non-decreasing objective function for randomly perturbed instances of, e.g., Shortest Path, Spanning Tree, and Perfect Matching.