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Another constraint considered in literature is a bound D on the length of a vehicle tour, under the objective of minimizing the number of routes. This is the Distance Constrained Vehicle Routing Problem (DVRP). It was raised and studied for applications in [7] and [8]. Routing problems like the DVRP can be directly encoded as instances of Minimum Set Cover, and thus often admit logarithmic approximations. The authors of [9] give a careful analysis of the set cover integer programming formulation of the DVRP and bound its integrality gap by O(log D) on general graphs and by O(1) on a tree.

Fd with the number of terminals of each Fi in [β, 3β), 1 ≤ i ≤ d. 3 An O(log n)-Approximation for the (k, 2)-Subgraph Problem In this section we prove Theorem 3. In fact (similar to the algorithm in [14]) our algorithm works for a slightly more general case in which along with the weighted graph G = (V, E) and integer k we are also given a set of terminals T ⊆ V and the goal is to find a minimum cost 2-edge-connected subgraph that contains at least k terminals. Our algorithm will round an LP relaxation directly instead of iteratively finding good density partial solutions as done in [14].

First we provide the details of the steps of the algorithm. Suppose L is the kth smallest d2 (v, r) value. L. We can start with L as our guess for opt and if the algorithm fails to return a feasible solution of cost at most O(opt · log n) then we double our guess opt and run the algorithm again. R. R. Salavatipour Let (x∗ , y ∗ ) be an optimum feasible solution to LP-k2EC with value opt∗ . For Step 5 of K2EC we round y values of the LP following the schema in [3]. The proof of following lemma is very similar to Lemma 3.

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