On the L∞-norm of extreme points for crossing supermodular directed network LPs

We discuss extensions of Jain’s framework for network design [8] that go beyond undirected graphs. The main problem is approximating a minimum cost set of directed edges that covers a crossing supermodular function. We show that iterated rounding gives a factor 3 approximation, where factor 4 was previously known and factor 2 was conjectured. Our bound is tight for the simplest interpretation of iterated rounding. We also show that (the simplest version of) iterated rounding has unbounded approximation ratio when the problem is extended to mixed graphs.