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

Abstract

Letw _i ∶V ×V →Q,i = 1, 2 be two weight functions on the possible edges of a directed or undirected graph with vertex setV such that for the cut function, the inequality

$$\delta _{w_2 } (T): = \sum\limits_{\scriptstyle i \in T \hfill \atop \scriptstyle j \notin T \hfill} {w_2 (ij) \ge 0.} $$

holds for everyT ε 2^V We consider the computation of the value λ(W ₁,W ₂,K) defined by

$$\tilde \lambda (w_{\text{1}} ,w_2 ,k): = \min \{ l|\delta _{w_1 } (T) + l\delta _{w_2 } (T) \geqslant k \forall \emptyset \subset T \subset V\} .$$

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 networkN = (V, E, c) with a rational valued capacity functionc∶ V ×V →Q ₊, and letk be a positive, rational number. Consider the problem of finding a capacity functionc′∶V ×V →Q ₊ such that, in the resulting networkN′ = (V, E, c + c′) the edge connectivity number λ_c+c′, is at leastk, and the maximal increasec′(ij) is minimal.

We give an algorithm which computes such an augmentation in strongly polynomial time.