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
Random pseudo-polynomial algorithms for exact matroid problems
Journal of Algorithms
Approximation schemes for the restricted shortest path problem
Mathematics of Operations Research
Many birds with one stone: multi-objective approximation algorithms
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The Constrained Minimum Spanning Tree Problem (Extended Abstract)
SWAT '96 Proceedings of the 5th Scandinavian Workshop on Algorithm Theory
Bicriteria Network Design Problems
ICALP '95 Proceedings of the 22nd International Colloquium on Automata, Languages and Programming
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
Rapid rumor ramification: approximating the minimum broadcast time
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Combinatorial Optimization: Theory and Algorithms
Combinatorial Optimization: Theory and Algorithms
Budgeted matching and budgeted matroid intersection via the gasoline puzzle
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
Matching based augmentations for approximating connectivity problems
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
A simple efficient approximation scheme for the restricted shortest path problem
Operations Research Letters
ESA'11 Proceedings of the 19th European conference on Algorithms
Multi-budgeted matchings and matroid intersection via dependent rounding
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Matroid and knapsack center problems
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
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A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Some classical optimization problems, such as spanning tree and forest, shortest path, (perfect) matching, independent set (basis) in a matroid or in the intersection of two matroids, become NP-hard even with one budget constraint. Still, for most of these problems efficient deterministic and randomized approximation schemes are known. For two or more budgets, typically only multi-criteria approximation schemes are available, which return slightly infeasible solutions. Not much is known however for strict budget constraints: filling this gap is the main goal of this paper. It is not hard to see that the above-mentioned problems whose solution sets do not correspond to independence systems are inapproximable already for two budget constraints. For the remaining problems, we present approximation schemes for a constant number k of budget constraints using a variety of techniques: i) we present a simple and powerful mechanism to transform multi-criteria approximation schemes into pure approximation schemes. This leads to deterministic and randomized approximation schemes for various of the above-mentioned problems; ii) we show that points in low-dimensional faces of any matroid polytope are almost integral, an interesting result on its own. This gives a deterministic approximation scheme for k-budgeted matroid independent set; iii) we present a deterministic approximation scheme for 2-budgeted matching. The backbone of this result is a purely topological property of curves in R2.