The maximum concurrent flow problem (MCFP) is a multicommodity flow problem in which every pair of entities can send and receive flow concurrently. The ratio of the flow supplied between a pair of entities to the predefined demand for that pair is called throughput and must be the same for all pairs of entities for a concurrent flow. The MCFP objective is to maximize the throughput, subject to the capacity constraints. We develop a fully polynomial-time approximation scheme for the MCFP for the case of arbitrary demands and uniform capacity. Computational results are presented. It is shown that the problem of associating costs (distances) to the edges so as to maximize the minimum-cost of routing the concurrent flow is the dual of the MCFP. A path-cut type duality theorem to expose the combinatorial structure of the MCFP is also derived. Our duality theorems are proved constructively for arbitrary demands and uniform capacity using the algorithm. Applications include packet-switched networks [ 1, 4, 81, and cluster analysis [ 161.

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The bipartite crossing number problem is studied and a connection between this problem and the linear arrangement problem is established. A lower bound and an upper bound for the optimal number of crossings are derived, where the main terms are the optimal arrangement values. Two polynomial time approximation algorithms for the bipartite crossing number are obtained. The performance guarantees are O(log n) and O(log 2 n) times the optimal, respectively, for a large class of bipartite graphs on n vertices. No polynomial time approximation algorithm which could generate a provably good solution had been known. For a tree, a formula is derived that expresses the optimal number of crossings in terms of the optimal value of the linear arrangement and the degrees, resulting in an O(n 1:6) time algorithm for computing the bipartite crossing number. The problem of computing a maximum weight biplanar subgraph of an acyclic graph is also studied and a linear time algorithm for solving it is derived. No polynomial time algorithm for this problem was known, and the unweighted version of the problem had been known to be NP-hard, even for planar bipartite graphs of degree at most 3.