Fractional Packing and Parametric Search Frameworks

Abstract

The following chapter introduces three generic frameworks that will be used throughout this thesis. On the one hand, the focus will be set on the parametric search framework due to Megiddo (1979, 1983), which can often be used to solve parametric variants of known combinatorial problems in strongly polynomial time. On the other hand, we will review the fractional packing framework by Garg and Koenemann (2007), which yields generic fully polynomial-time approximation schemes (FPTASs) for problems that can be formulated as packing LPs. In the second part of this chapter, we combine both approaches into a generalized packing framework that may be used to derive FPTASs for packing problems on finitely generated polyhedral cones. In particular, for a given oracle for this cone, the result extends to the case of cones that are generated by an exponential number of vectors. We show that we obtain FPTASs with varying time complexities for oracles with varying power. Finally, we show that this generalized packing framework, which will be used in Chapter 4 and 6, yields FPTASs for a large class of network flow problems in general.