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

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 whenthe 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. Definitionscont. • 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. Definitionscont. • Graph is k-vertex-connectedif 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.

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. More result cont. • 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 .

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. More Definitions • 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.

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 ofif 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