Even though we mentioned the paper by Jain [75] as the first explicit application of the iterative method to approximation algorithms, several earlier results can be reinterpreted in this light, which is what we set out to do in this chapter. We will first present a result by Beck and Fiala [12] on hypergraph discrepancy, whose proof is closest to other proofs in this book. Then we will present a result by Steinitz [127] on rearrangements of sums in a geometric setting, which is the earliest application that we know of. Then we will present an approximation algorithm by Skutella [123] for the single source unsplittable flow problem. Then we present the additive approximation algorithm for the bin packing problem by Karmarkar and Karp [77], which is still one of the most sophisticated uses of the iterative relaxation method. Finally, we sketch a recent application of the iterative method augmented with randomized rounding to the undirected Steiner tree problem [20] following the simplification due to Chakrabarty et al. [24].

A discrepancy theorem

In this section, we present the Beck–Fiala theorem from discrepancy theory using an iterative method. Given a hypergraph G = (V, E), a 2-coloring of the hypergraph is defined as an assignment ψ: V → {-1, +1} on the vertices. The discrepancy of a hyperedge e is defined as discχ(e) = ∑v∈e ψ(v), and the discrepancy of the hypergraph G is defined as discψ(G) = maxe∈E(G) |{discψ(e)}|.