On balanced edge connectivity and applications to some bottleneck augmentation problems in networks

Let w i : V V ! Q, i = 1; 2 be two weight functions on the possible edges of a directed or undirected graph with vertex set V such that for the cut function, the inequality w2 (T) := X i2T j6 2T w 2 (ij) 0; holds for every T 2 2 V. We consider the computation of the value ~ (w 1 ; w 2 ; k) deened by ~ (w 1 ; w 2 ; k) := minfl j w1 (T) + l w2 (T) k 8 ; T V g : We show that the associated decision problem is NP-complete, but for a class of instances we can give a polynomial time algorithm. This class is closely related to the following bottleneck augmentation problem. Consider a network N = (V; E; c) with a rational valued capacity function c: V V ! Q + , and let k be a positive, rational number. Consider the problem of nding a capacity function c 0 : V V ! Q + such that, in the resulting network N 0 = (V; E; c + c 0) the edge connectivity number c+c 0 is at least k, and the maximal increase c 0 (ij) is minimal. We give an algorithm which computes such an augmentation in strongly polynomial time.