Algorithms for a network design problem with crossing supermodular demands

We present approximation algorithms for a class of directed network design problems. The network design problem is to find the minimum cost subgraph such that for each vertex set S there are at least f(S) arcs leaving the set S. In the last 10 years general techniques have been developed for designing approximation algorithms for undirected network design problems. Recently, Kamal Jain gave a 2-approximation algorithm for the case when the function f is weakly supermodular. There has been very little progress made on directed network design problems. The main techniques used for the undirected problems do not have simple extensions to the directed case. András Frank has shown that in a special case when the function f is intersecting supermodular the problem can be solved optimally. In this article, we use this result to get a 2-approximation algorithm for a more general case when f is crossing supermodular. We also extend Jain's techniques to directed problems. We prove that if the function f is crossing supermodular, then any basic solution of the LP relaxation of our problem contains at least one variable with value greater than or equal to ¼. This result implies a 4-approximation algorithm for the class of directed network design problems.