1. A 2-Approximation algorithm for finding an optimum 3-Vertex-Connected Spanning Subgraph

2. The problem Having sites, the problem is to build roads between them, so you'll be able to travel from each city to the other. You have to build the roads in a way that even if sites are destroyed, the other cities will be still connected. You’ll need to build it as cheap as possible when t he cost of each road (if possible to build) is known.

3. Definitions Graph, Directed graph degree of v Spanning subgraph of graph Weighted Graph Weight of an edge set

4. Definitions cont. A graph is connected if for any two vertices of there is a path connecting them. A subset is a vertex cut of if is disconnected. If then is called a k-cut A side of a cut is the vertex set of a connected component of.

5. Graph is k-vertex-connected if either it is a complete graph of vertices or if it has at least vertices and contains no with. Alternatively it can be said that is k- vertex-connected graph if for every set of vertices, and G\V’ is connected. Connectivity of G, defined to be the maximum k, for which G is k-connected. Definitions cont.

6. k-connected subgraph problem is an NP- hard. This is our motivation to find approximation algorithms. Approximation Algorithm is called if it is a polynomial time algorithm and produces a solution of weight no more than times the weight of the optimum solution.

7. Known Approximation Ravi and Williamson [1997]: For an arbitrary k, it achieves where For k=2 achieves 3-approximation k=3 achieves -approximation k=4 achieves -approximation

8. More results Improved results for particular cases: In case of edge weights satisfying triangle inequality, Kuller-Raghavachari [1996] suggested an –approximation, algorithm for an arbitrary k.

9. More result (cont.) Cheyiyan-Thurimella [1996]: -approximation for finding minimum size k-connected spanning subgraph, for an arbitrary k, meaning finding the k-connected spanning subgraph with minimal number of edges.

10. Kuller-Raghavachari [1996]: achieves a for. Result was improved by Penn and Shasha-Krupnik [1997] to for. Penn and Shasha-Krupnik [1997] also introduced a for. More result cont.

11. Today we show an improved result: a 2-approximation algorithm for finding a minimim weight 3-connected subgraph, introduced by Auletta, Dinitz,Nutov and Parente [1999]

12. Path are internally disjoint paths if no two of them have an internal vertex in common. Menger’s therom: For any graph G and its vertices s,t holds: The minimal size if a cut separating t from s equals the maximum number of vertex-disjoint paths between s and t. More Definitions

13. Definition: Graph is k-out-connected from vertex r if there exists k internally vertex- disjoint simple paths to every other vertex If there are 2 vertices with k internally disjoint simple paths between them, then for every implies.

14. Corollary: A graph is a k-out-connected from vertex r if it has no with separating r from some other vertex

15. Conclusion: In k-out-connected graph from vertex r, any with, if exists, must contain.

16. What is the motivation to use k-out- connected graph algorithm for the problem of finding minimum weight k-connected subgraph? There is known algorithm by Frank and Tardos [1996] that find in a directed graph a minimum weight k-out-connected subdigraph in polynomial time.

17. Lemma 1: Let be a k-out-connected graph from, and let be an of with. Then and for any side holds:.

18. Corollary: Also exists that for the above graph

19. Corollary: Let be a k-out-connected graph from a vertex r of degree. Then is. In particular if then is. Conclusion: For such a vertex r (of degree k) k- connected graph and k-out-connected graph from r are equal.

20. Graph G=(V,E) is given and its weight function,. D(G)= weight digraph obtained from (G,w): Each undirected edge is replaced by 2 directed edge with same weight as the undirected edge. U(D) underlying graph of digraph D, where for each directed edge replace it by undirected edge.

21. If is then it is from. If is undirected graph and from then is also a from.

22. Theorem [Halin]: Any minimally graph has a vertex of degree k. Corollary: It follows from Halin’s theorem that in any graph, exists a minimum weight subgraph with a vertex of degree. We denotes this vertex by.

23. Out Connected Subgraph Algorithm Input: A weighted graph, and an integer k. Output: a subgraph of and a vertex such that is k-out-connected from and if exists.

24. Set undefined, For every vertex do: (1) Set (2) Find a minimum weight k-out- connected from r subdigraph of if such exists. (3) If the degree of r in is k and then set:

25. Remarks: The algorithm finds k-out-connected subdigraphs with a minimal outdegree in r. From all those subdigraph it chooses the subdigraph with the minimal weight.

26. Lemma: For any integer and for any weighted graph G that contains a spanning subgraph which is k-out- connected from a vertex of degree k, the algorithm outputs such a subgraph of weight at most twice the minimal possible. The complexity is.

27. Theorem: For any and any weighted k-connected graph G, algorithm outputs a spanning subgraph of G of weight at most in time where is the weight of the optimal subgraph of Theorem: For OCSA is a 2-approximation algorithm for the minimum weight k-connected subgraph problem with complexity