Efficient probabilistically checkable proofs and applications to approximations
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
The Complexity of Multiterminal Cuts
SIAM Journal on Computing
PCP characterizations of NP: towards a polynomially-small error-probability
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
On the power of unique 2-prover 1-round games
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications
SIAM Journal on Computing
Improved results for directed multicut
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Hardness of Approximating Problems on Cubic Graphs
CIAC '97 Proceedings of the Third Italian Conference on Algorithms and Complexity
Relations between Average Case Complexity and Approximation Complexity
CCC '02 Proceedings of the 17th IEEE Annual Conference on Computational Complexity
Expander flows, geometric embeddings and graph partitioning
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Euclidean distortion and the sparsest cut
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Approximating Directed Multicuts
Combinatorica
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Directed metrics and directed graph partitioning problems
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Logarithmic hardness of the directed congestion minimization problem
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Hardness of cut problems in directed graphs
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
An O(√n)-approximation algorithm for directed sparsest cut
Information Processing Letters
ON THE HARDNESS OF APPROXIMATING MULTICUT AND SPARSEST-CUT
Computational Complexity
Hardness of routing with congestion in directed graphs
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Improved approximation for directed cut problems
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Fixed-parameter tractability of multicut parameterized by the size of the cutset
Proceedings of the forty-third annual ACM symposium on Theory of computing
Multicommodity flows and cuts in polymatroidal networks
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Theoretical Computer Science
Sparsest cut on bounded treewidth graphs: algorithms and hardness results
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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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 the number 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, while the standard 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 largest gap, achievable for any graph, between the maximum flow value and the minimum cost solution for the corresponding cut problem. 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 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 flow-cut 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 2Ω(log1−&epsis; n) for any constant &epsis; 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.