In this paper, we study the k-forest problem in the model of resource
augmentation. In the k-forest problem, given an edge-weighted graph G(V,E),
a parameter k, and a set of m demand pairs ⊆V×V, the
objective is to construct a minimum-cost subgraph that connects at least k
demands. The problem is hard to approximate---the best-known approximation
ratio is O(min{n,k}). Furthermore, k-forest is as hard to
approximate as the notoriously-hard densest k-subgraph problem.
While the k-forest problem is hard to approximate in the worst-case, we
show that with the use of resource augmentation, we can efficiently approximate
it up to a constant factor.
First, we restate the problem in terms of the number of demands that are {\em
not} connected. In particular, the objective of the k-forest problem can be
viewed as to remove at most m−k demands and find a minimum-cost subgraph that
connects the remaining demands. We use this perspective of the problem to
explain the performance of our algorithm (in terms of the augmentation) in a
more intuitive way.
Specifically, we present a polynomial-time algorithm for the k-forest
problem that, for every ϵ>0, removes at most m−k demands and has
cost no more than O(1/ϵ2) times the cost of an optimal algorithm
that removes at most (1−ϵ)(m−k) demands