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
Efficiently computing succinct trade-off curves
Theoretical Computer Science - Automata, languages and programming: Algorithms and complexity (ICALP-A 2004)
Small Approximate Pareto Sets for Biobjective Shortest Paths and Other Problems
SIAM Journal on Computing
Pareto optimal solutions for smoothed analysts
Proceedings of the forty-third annual ACM symposium on Theory of computing
High-Performance heuristics for optimization in stochastic traffic engineering problems
LSSC'09 Proceedings of the 7th international conference on Large-Scale Scientific Computing
Hi-index | 5.23 |
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 nonlinear) 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 the objective function, however, shows exponential growth then the decision-maker's problem is NP-hard to approximate within any polynomial 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.