Approximating some network-design problems with degree constraints

We study several network design problems with degree constraints. For the degree-constrained 2 vertex-connected subgraph problem we obtain a factor 6 violation for the degrees with 4 approximation for the cost. This improves upon the logarithmic degree violation and no cost guarantee obtained by Feder, Motwani, and Zhu (2006). Then we consider the problem of finding an arborescence with at least k terminals and with minimum maximum outdegree. We show that the natural LP-relaxation has a gap of Ω (√ k ) or Ω ( n ) with respect to the multiplicative degree bound violation. We overcome this hurdle by a combinatorial O( √ (k log k)/∆∗)-approximation algorithm, where ∆∗ denotes the maximum degree in the optimum solution. We also give an Ω(log n) lower bound on approximating this problem. Then we consider the undirected version of this problem, however, with an extra diameter constraint, and give an Ω(log n) lower bound on the approximability of this version. We consider a closely related prize-collecting degreeconstrained Steiner Network problem. We obtain several results in this direction by reducing the prize-collecting variant to the regular one. Finally, we consider the a degree constraint problem in which the degree bounds have hard capacity and a violation of this capacity is not allowed. We study the Steiner tree problem in which some of the terminals are required to be leaves. We show that this seemingly simple problem is equivalent to the Group Steiner problem in trees.