### Polynomial flow-cut gaps and hardness of directed cut problems

Julia Chuzhoy, Sanjeev Khanna
2009 Journal of the ACM
We study the multicut and the sparsest cut problems in directed graphs. In the multicut problem, we are a given an n-vertex graph G along with k source-sink pairs, and the goal is to find the minimum cardinality subset of edges whose removal 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. The
more » ... linear programming relaxation for multicut corresponds, by LP-duality, to the well-studied maximum (fractional) multicommodity flow problem, while the natural LP-relaxation for sparsest cut corresponds to maximum concurrent flow. Therefore, the integrality gap of the linear programming relaxation for multicut/sparsest cut is also the flow-cut gap: the maximum ratio, achievable for any graph, between the maximum flow value and the minimum cost solution for the corresponding cut problem. Starting with the celebrated max flow-min cut theorem of Ford and Fulkerson, flow-cut gaps have played a central role in combinatorial optimization. For many NP-hard network optimization problems, the best known approximation guarantee corresponds to our understanding of the appropriate flow-cut gap. Our first result is that the flow-cut gap between maximum multicommodity flow and minimum multicut isΩ(n 1/7 ) in directed graphs. We show a similar result for the gap between maximum concurrent flow and sparsest cut in directed graphs. These results improve upon a long-standing lower bound of Ω(log n) for both types of flow-cut gaps. We notice * that these polynomially large flow-cut gaps are in a sharp contrast to the undirected setting where both these flowcut gaps are known to be Θ(log n). Our second result is that both directed multicut and sparsest cut are hard to approximate to within a factor of 2 Ω(log 1− n) for any constant > 0, unless NP ⊆ ZPP. This improves upon the recent Ω(log n/ log log n)-hardness result for these problems. We also show that existence of PCP's for NP with perfect completeness, polynomially small soundness, and constant number of queries would imply a polynomial factor hardness of approximation for both these problems. All our results hold for directed acyclic graphs.