Polynomial flow-cut gaps and hardness of directed cut problems

Abstract

We study the multicut and the sparsest cut problems in directed graphs. In the multicut problem, we are a given an n-vertex graphG 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 natural linear programming relaxation for multicut corresponds, by LP-duality, to the well-studied maximum (fractional) multicommodity flow problem, whilethe 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-mincut theorem of Ford and Fulkerson, flow-cut gaps have played acentral 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 is ~Ω (n1/7) in directed graphs. We show a similar result for the gap between maximum concurrent flow and sparsest cutin directed graphs. These results improve upon a long-standing lowerbound of Ω(log n) for both types of flow-cut gaps. We notice that these polynomially large flow-cut gaps are in a sharp contrastto the undirected setting where both these flow-cut gaps are knownto be Θ(log n). Our second result is that both directed multicut and sparsest cut are hard to approximate to within a factor of 2Ω(log1-en) for any constant e>0,unless NP ⊆ ZPP. This improves upon the recent Ω(log n / log log n)-hardness result for these problems. We also showthat 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.