|
| | | |
Augmenting Edge-Connectivity between Vertex Subsets
Ishii, T. and Makino, K.
Given a graph G = (V,E) and a requirement function
r : W1 x W2 -> R+ for two families W1, W2 \subseteq 2V - {0}, we consider the problem (called area-toarea
edge-connectivity augmentation problem) of augmenting
G by a smallest number of new edges so that
the resulting graph �G satisfies \delta �G(X) � r(W1;W2)
for all X \subseteq V , W1 in W1, and W2 in W2 with
W1 \subseteq X \subseteq V - W2, where \delta G(X) denotes the degree
of a vertex set X in G. This problem can be
regarded as a natural generalization of the global, local,
and node-to-area edge-connectivity augmentation
problems.
In this paper, we show that there exists a
constant c such that the problem is inapproximable
within a ratio of c log \alpha(W1,W2), unless P=NP,
even restricted to the directed global node-to-area
edge-connectivity augmentation or undirected local
node-to-area edge-connectivity augmentation, where
\alpha(W1,W2) denotes the number of pairs W1 in W1 and
W2 in W2 with r(W1;W2) > 0. We also provide an
O(log \alpha(W1,W2))-approximation algorithm for the
area-to-area edge-connectivity augmentation problem.
This together with the negative result implies
that the problem is \Theta(log \alpha(W1,W2))-approximable,
unless P=NP, which solves open problems for nodeto-
area edge-connectivity augmentation.
Furthermore, we characterize the node-toarea
and area-to-area edge-connectivity augmentation
problems as the augmentation problems with
modulotone and extended modulotone functions.
| |