Hardness of cut problems in directed graphs

We study the approximability of the multicut and the (non-bipartite) sparsest cut problems in directed graphs. In the multicut problem, we are a given a graph G along with k source-sink pairs, and the goal is to find a smallest subset of edges whose deletion separates all source-sink pairs. The sparsest cut problem has the same input, but the goal is to find a subset of edges to delete so as to minimize the ratio of deleted edges to the number of source-sink pairs that are separated by this deletion. Study of algorithms for cut problems is intimately connected to the dual notion of flows in networks, and many approximation algorithms for cut problems use a flow solution as a starting point. The best known approximation algorithm for directed multicut is based on this approach and gives an O(√n)-approximation. On the other hand, the gap between the maximum multicommodity flow and the minimum multicut is known to be Ω(min(k , log n)). While this flow-cut gap may be interpreted as an evidence of inherent difficulty in designing good approximation algorithms for directed multicut, the strongest hardness result known is an APX-hardness. Even assuming the Unique Games Conjecture, only an ω(1)-hardness is known. Similar bounds hold for the directed sparsest cut problem.Our main result is that directed multicut is Ω(log n / log log n)-hard to approximate unless NP ⊆ DTIME (npolylog n). We show that this hardness result holds even when we allow a bicriteria relaxation, where the approximate solution is required to separate only a constant fraction of the pairs. This bicriteria hardness allows us to infer an Ω(log n / log log n)-hardness for the directed (non-bipartite) sparsest cut problem.