Iterative Rounding Approximation Algorithms for Degree-Bounded Node-Connectivity Network Design

Abstract

Abstract. We consider the problem of finding a minimum edge cost subgraph of a graph satisfying both given node-connectivity requirements and degree upper bounds on nodes. We present an iterative rounding algorithm of the biset LP relaxation for this problem. For directed graphs and k-out-connectivity requirements from a root, our algorithm computes a solution that is a 2-approximation on the cost, and the degree of each node v in the solution is at most 2b(v) + O(k) where b(v) is the degree upper bound on v. For undirected graphs and element-connectivity requirements with maximum connectivity requirement k, our algorithm computes a solution that is a 4-approximation on the cost, and the degree of each node v in the solution is at most 4b(v) + O(k). These ratios improve the previous O(log k)-approximation on the cost and O(2 k b(v)) approximation on the degrees. Our algorithms can be used to improve approximation ratios for other nodeconnectivity problems such as undirected k-out-connectivity, directed and undirected k-connectivity, and undirected rooted k-connectivity and subset k-connectivity.

Improved approximation algorithms for degree-bounded network design problems with node connectivity requirements

“…approximation for k-Connected Subgraph; the latter result holds for all values of k. In joint work with Nutov [7], the authors of [8] strengthened their analysis and obtained an (O(1), O(1)b(v) + O(k)) approximation for Elem-SNDP. By combining the approach of Cheriyan and Vegh [2] for the k-Connected Subgraph problem without degree bounds with the algorithm of [8] for the degree bounded Rooted k-Connectivity problem and the algorithm of [7] for the degree bounded Elem-SNDP problem, one obtains an (O(1), O(1)b(v) + O(k)) approximation for the degree bounded k-Connected Subgraph problem when the number of nodes is at least (k − 1) 3 − k.…”

“…Recently, Nutov [18] considered several degree bounded network design problems with element and vertex connectivity requirements. Nutov showed that one can combine the augmentation framework of Williamson et al [20] with iterated rounding techniques in order to construct a solution that violates the degree bounds by a multiplicative factor that is exponential in k. Some of Nutov's results include an (O(log k), O(2 k )b(v)) approximation for Elem-SNDP and Rooted k-Connectivity, and an (O(k), O(2 k )b(v)) approximation for k-Connected Subgraph; the latter result holds for all values of k. Shortly after, Fukunaga and Ravi [8] significantly improved the approximation guarantees for these problems; their algorithms are based on iterated rounding and their analyses use a novel technical insight. The results of [8]…”

“…Our work closes the gap between several edge connectivity problems and their element or vertex counterparts by providing approximation algorithms that nearly match the best guarantees for the edge connectivity setting. As it is common in this area, we use iterated rounding techniques but our analyses are a departure from the approach of Fukunaga and Ravi [8] and Fukunaga, Nutov, and Ravi [7]. Our main technical insight is inspired by the work of Nutov [18] and it takes advantage of the special structure of the SNDP problem.…”

“…Table 1 summarizes our main results and the best approximations for the problems we consider that were given in previous work. We refer the reader to [8] for additional results and references on degree bounded network design problems with element and vertex connectivity requirements.…”

“…We remark that these algorithms lead to degree violations that are exponential in the maximum requirement k of any pair. Shortly after, Fukunaga et al [8,7] significantly improved the best guarantees for these problems in which the degree violations are only polynomial in k.…”