No abstract

The following problem is considered: Given an integer K , a graph G with two distinct vertices s and t , find the maximum number of disjoint paths of length K from s to t . The problem has several variants: the paths may be vertex-disjoint or edge-disjoint, the lengths of the paths may be equal to K or bounded by K , the graph may be undirected or directed. It is shown that except for small values of K all the problems are NPcomplete. Assuming P f NP, for each problem, the largest value of K for which the problem is not NP-complete is found. Whenever a polynomial algorithm exists, an efficient algorithm is described.

No abstract

Efficient algorithms for finding maximum flow in planar networks are presented. These algorithms take advantage of the planarity and are superior to the most efficient algorithms to date, If the source and the terminal are on the same face, an algorithm of Berge is improved and its time complexity is reduced to O(n log n). In the general case, for a given D > 0 a flow of value D is found if one exists; otherwise, it is indicated thatno such flow exists. This algorithm requires O(n 2 log n) time. If the network is undirected a minimum cut may be found in O(n log n) time. All algorithms require O(n)space.

No abstract