Approximating Minimum-Cost k-Node Connected Subgraphs via Independence-Free Graphs

We present a 6-approximation algorithm for the minimum-cost $k$-node connected spanning subgraph problem, assuming that the number of nodes is at least $k^3(k-1)+k$. We apply a combinatorial preprocessing, based on the Frank-Tardos algorithm for $k$-outconnectivity, to transform any input into an instance such that the iterative rounding method gives a 2-approximation guarantee. This is the first constant-factor approximation algorithm even in the asymptotic setting of the problem, that is, the restriction to instances where the number of nodes is lower bounded by a function of $k$.Comment: 20 pages, 1 figure, 28 reference

On Generalizations of Network Design Problems with Degree Bounds

Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely, laminar crossing spanning tree), and (2) by incorporating `degree bounds' in other combinatorial optimization problems such as matroid intersection and lattice polyhedra. We give new or improved approximation algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure

Technology diffusion in communication networks

The deployment of new technologies in the Internet is notoriously difficult, as evidence by the myriad of well-developed networking technologies that still have not seen widespread adoption (e.g., secure routing, IPv6, etc.) A key hurdle is the fact that the Internet lacks a centralized authority that can mandate the deployment of a new technology. Instead, the Internet consists of thousands of nodes, each controlled by an autonomous, profit-seeking firm, that will deploy a new networking technology only if it obtains sufficient local utility by doing so. For the technologies we study here, local utility depends on the set of nodes that can be reached by traversing paths consisting only of nodes that have already deployed the new technology. To understand technology diffusion in the Internet, we propose a new model inspired by work on the spread of influence in social networks. Unlike traditional models, where a node's utility depends only its immediate neighbors, in our model, a node can be influenced by the actions of remote nodes. Specifically, we assume node v activates (i.e. deploys the new technology) when it is adjacent to a sufficiently large connected component in the subgraph induced by the set of active nodes; namely, of size exceeding node v's threshold value \theta(v). We are interested in the problem of choosing the right seedset of nodes to activate initially, so that the rest of the nodes in the network have sufficient local utility to follow suit. We take the graph and thresholds values as input to our problem. We show that our problem is both NP-hard and does not admit an (1-o(1) ln|V| approximation on general graphs. Then, we restrict our study to technology diffusion problems where (a) maximum distance between any pair of nodes in the graph is r, and (b) there are at most \ell possible threshold values. Our set of restrictions is quite natural, given that (a) the Internet graph has constant diameter, and (b) the fact that limiting the granularity of the threshold values makes sense given the difficulty in obtaining empirical data that parameterizes deployment costs and benefits. We present algorithm that obtains a solution with guaranteed approximation rate of O(r^2 \ell \log|V|) which is asymptotically optimal, given our hardness results. Our approximation algorithm is a linear-programming relaxation of an 0-1 integer program along with a novel randomized rounding scheme.National Science Foundation (S-1017907, CCF-0915922