The NP-completeness column: An ongoing guide
Journal of Algorithms
Finding good approximate vertex and edge partitions is NP-hard
Information Processing Letters
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
A Polylogarithmic Approximation of the Minimum Bisection
SIAM Journal on Computing
Which Problems Have Strongly Exponential Complexity?
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Finding small balanced separators
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Discrete Applied Mathematics
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A bisection of an n-vertex graph is a partition of its vertices into two sets S and T, each of size n/2. The bisection cost is the number of edges connecting the two sets. In directed graphs, the cost is the number of arcs going from S to T. Finding a minimum cost bisection is NP-hard for both undirected and directed graphs. For the undirected case, an approximation of ratio O(log2n) is known. We show that directed minimum bisection is not approximable at all. More specifically, we show that it is NP-hard to tell whether there exists a directed bisection of cost 0, which we call oneway bisection. In addition, we study the complexity of the problem when some slackness in the size of S is allowed, namely, (1/2 - ε)n ≤ |S| ≤ (1/2 + ε)n. We show that the problem is solvable in polynomial time when ε = Ω(1/logn), and provide evidence that the problem is not solvable in polynomial time when ε = o(1/(logn)4).