Approximating k-Forest with Resource Augmentation: A Primal-Dual Approach

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 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/ε) times the cost of an optimal algorithm that removes at most (1− ε)(m− k) demands.