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The Fault-Tolerant Group Steiner Problem

The Fault-Tolerant Group Steiner Problem. Rohit Khandekar IBM Watson Joint work with G. Kortsarz and Z. Nutov. Fault-tolerant group Steiner problem. Given: A weighted graph G(V,E) , a collection of subsets (groups) g i  V and a root r .

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The Fault-Tolerant Group Steiner Problem

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  1. The Fault-Tolerant GroupSteiner Problem Rohit Khandekar IBM Watson Joint work with G. Kortsarz and Z. Nutov

  2. Fault-tolerant group Steiner problem • Given: A weighted graph G(V,E), a collection of subsets (groups) gi  V and a root r. • The goal: Find a minimum weight subgraph in which for eachgi, at least 2 vertices have edge (or vertex) disjoint paths to r.

  3. An example r g3 g1 g2 g1 g2 g3

  4. Previous work on fault-tolerant problems: Steiner networks Steiner Network: Instance: A complete graph with edge (or vertex) costs, connectivity requirements r(u,v) Objective: Min-cost subgraph with r(u,v) edge (or vertex) disjoint uv-paths for all u,v in V k-edge-Connected Subgraph:r(u,v) = k for all u,v, edge-disjointness k-vertex-Connected Subgraph:r(u,v) = k for all u,v, vertex-disjointness

  5. Previous work on Steiner Network Edge case: A sequence of papers reaching a 2-approximation [Jain 98] Vertex case: • Labelcover hard [Kortsarz, Krauthgamer, Lee 04] • kε approximation is unlikely for some universal ε>0[Chakraborty, Chuzhoy, Khanna 08] • Undirected and directed problems are equivalent for k>n/2 [Lando, Nutov 08] • O(log n)-approximation for metric cost [Cheriyan, Vetta 05] • O(k3 log n)-approximation [Chuzhoy, Khanna 09]

  6. 2-connectivity problems (like ours) • 2-edge-connected subgraph spanning k vertices: O(log n log k)[Lau, Naor, Salavatipour, Singh 09] (fault tolerant version of k-MST) • Same problem with 2-vertex-connectivity: O(log n log k)[Chekuri, Korula 08] • Finding buy at bulk trees with 2-vertex-disjoint paths from the terminals to the root: O(log3 n)[Antonakopoulos, Chekuri, Shepherd, Zhang 07]

  7. Our results FTGS = Fault tolerant group Steiner, GS = Group Steiner, VC = Vertex cover

  8. Our results FTGS = Fault tolerant group Steiner, GS = Group Steiner, VC = Vertex cover

  9. Why is our problem difficult? • Known algorithms for Group Steiner tree are based on approximating the given metric by tree metrics [Bartal 98], [Fakcharoenphol, Rao, Talwar 03] and solving the problem on trees. • This reduction does not preserve the connectivity information and hence cannot be used here. • An intriguing question: Can we approximate Group Steiner problem without first transforming the graph into a tree?

  10. Algorithm for FTGS-2 (edge case) • As |gi| = 2, all terminals must be connected to the root in any feasible solution. • Therefore we first find a STEINER TREE T connecting all terminals to the root (1.55-approximation). • Then we augment T to a feasible FTGS-2 solution.

  11. Violated sets • Say that X  V is violated if there is only one edge leaving X, but there should be two edges leaving X (i.e., X does not contain r but contains a group). • Claim: If X and Y are violated, either X U Y and X ∩ Y are both violated, or X-Y and Y-X are both violated. • Such a family of violated sets is called “uncrossable”.

  12. Why are violated sets uncrossable? • For any violated set X, the set X ∩ T must be a sub-tree of T containing an entire group. • Subtrees are laminar! (i.e., either two subtrees are disjoint or one is contained in the other.) gi X ∩ T

  13. g1 g2 X ∩ T Y ∩ T Y ∩ T g2 g1 X ∩ T The two cases X-Y=X and Y-X=Y X∩Y=Y and XUY=X

  14. Consequence • The problem of finding a minimum cost cover of an uncrossable family admits 2 approximation (Primal-Dual)[Goemans, Goldberg, Plotkin, Shmoys, Tardos 94]. • Therefore, overall we get 1.55 + 2 = 3.55 approximation. • It is also easy to see that the problem is Vertex Cover hard.

  15. Algorithm for FTGS-2 (vertex case) • First step: Steiner tree (same) • Second step: Augmentation problem is now different u u2 u1 g

  16. The augmentation problem Theorem: The group g is satisfied iff either u1 or u2 is 2-vertex-connected with r. u u2 u1 g

  17. The augmentation problem r u u2 u1 g

  18. The augmentation problem Profit(v) = number of groups g for which v serves the role of either u1 or u2 u u2 u1 g

  19. Density version of 2-vertex-connected graph problem • Given a graph with profits on vertices, find a subgraph H that minimizes the ratio of cost(H) to the profit of vertices that are 2-vertex-connected to r in H. • O(log n)-approximation [Chekuri, Korula 08] • This combined with the set-cover analysis gives O(log2 n)-approximation.

  20. FTGS-k (|g| ≤ k for all groups g) • A similar argument with a careful counting gives O(k log2 n) approximation if groups are assumed to be disjoint.

  21. Thanks!

  22. How many groups can u1 or u2 cover? r P=4 P=3 g2 P=2 P=1 P=1 g2 P=2 g1 g3 g1 g1 g3 g4 g3 g4 g3

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