Approximation Schemes for Multi-Budgeted Independence Systems ⋆ Fabrizio Grandoni 1 and Rico Zenklusen 2⋆⋆ 1 Computer Science Department, University of Rome Tor Vergata, grandoni@disp.uniroma2.it 2 Department of Mathematics, EPFL, rico.zenklusen@epfl.ch Abstract. 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 bud- gets, 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 al- ready 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 mech- anism to transform multi-criteria approximation schemes into pure ap- proximation schemes. This leads to deterministic and randomized ap- proximation 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 de- terministic 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 R 2 . 1 Introduction In many applications, one has to compromise between several, partially conflict- ing goals. Multi-Objective Optimization is a broad area of study in Operations Research, Economics and Computer Science (see [8, 22] and references therein). A variety of approaches have been employed to formulate such problems. Here we adopt the Multi-Budgeted Optimization approach [22]: we cast one of the goals ⋆ Partially developed while the first author was visiting EPFL. ⋆⋆ Partially supported by the Swiss National Science Foundation, grant number: PBEZP2-129524.