The k-forest problem is a common generalization of both the k-MST and the dense-ksubgraph problems. Formally, given a metric space on n vertices V, with m demand pairs ⊆ V × V and a “target ” k ≤ m, the goal is to find a minimum cost subgraph that connects at least k pairs. In this paper, we give an O(min { √ n · log k, √ k})-approximation algorithm for k-forest, improving on the previous best ratio of O(min{n 2/3, √ m} log n) by Segev and Segev. We then apply our algorithm for k-forest to obtain approximation algorithms for several Dial-a-Ride problems. The basic Dial-a-Ride problem is the following: given an n point metric space with m objects each with its own source and destination, and a vehicle capable of carrying at most k objects at any time, find the minimum length tour that uses this vehicle to move each object from its source to destination. We want that the tour be non-preemptive: that is, each object, once picked up at its source,

An instance of the k-generalized connectivity problem consists of an undirected graph G = (V, E), whose edges are associated with non-negative costs, and a collection D = {(S1, T1),..., (Sd, Td)} of distinct demands, each of which comprises a pair of disjoint vertex sets. We say that a subgraph H ⊆ G connects a demand (Si, Ti) when it contains a path with one endpoint in Si and the other in Ti. Given an integer parameter k, the goal is to identify a minimum cost subgraph that connects at least k demands in D. Alon, Awerbuch, Azar, Buchbinder and Naor (SODA ’04) seem to have been the first to consider the generalized connectivity paradigm as a unified machinery for incorporating multiplechoice decisions into network formation settings. Their main contribution in this context was to devise a multiplicative-update online algorithm for computing log-competitive fractional solutions, and to propose provably-good rounding procedures for important special cases. Nevertheless, approximating the generalized connectivity problem in its unconfined form, where one makes no structural assumptions about the underlying graph and collection of demands, has remained an open question up until a recent O(log 2 n log 2 d) approximation due to Chekuri, Even, Gupta and Segev (SODA ’08). Unfortunately, the latter result does not extend to connecting a pre-specified number of demands. Furthermore, even the simpler case of singleton demands has been established as a challenging computational task, when Hajiaghayi and Jain (SODA ’06) related its inapproximability to that of dense k-subgraph. In this paper, we present the first non-trivial approximation algorithm for k-generalized connectivity, which is derived by synthesizing several techniques originating in probabilistic embeddings of finite metrics, network design, and randomization. Specifically, our algorithm constructs, with constant probability, a feasible subgraph whose cost is within a factor of O(n 2/3 · polylog(n, k)) of optimal. We believe that the fundamental approach illustrated in the current writing is of independent interest, and may be applicable in other settings as well.