In previous chapters we have seen the definition of a constant factor approximation algorithm. In this chapter, we will introduce the notion of a polynomial time approximation scheme (PTAS), which allows approximability to any required degree. To illustrate how PTAS works, we will study two examples, including the knapsack problem and the bin packing problem. The dynamic programming technique w...

We consider the graph k-partitioning problem under min-max objective, termed as \(\textsc {Minmax}\;k\text {-}\textsc {cut}\). The input here is a \(G=(V,E)\) with non-negative edge weights \(w:E\rightarrow \mathbb {R}_+\) and an integer \(k\ge 2\) goal to partition vertices into k non-empty parts \(V_1, \ldots , V_k\) so minimize \(\max _{i=1}^k w(\delta (V_i))\). Although minimizing sum objec...