Скачать презентацию The Fault-Tolerant Group Steiner Problem Rohit Khandekar IBM Скачать презентацию The Fault-Tolerant Group Steiner Problem Rohit Khandekar IBM

3dc988d47340f3cf05d4b5441173082e.ppt

  • Количество слайдов: 22

The Fault-Tolerant Group Steiner Problem Rohit Khandekar IBM Watson Joint work with G. Kortsarz 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 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 each gi, at least 2 vertices have edge (or vertex) disjoint paths to r.

An example r g 1 g 2 g 3 g 1 g 2 An example r g 1 g 2 g 3 g 1 g 2

Previous work on fault-tolerant problems: Steiner networks Steiner Network: Instance: A complete graph with 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

Previous work on Steiner Network Edge case: A sequence of papers reaching a 2 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(k 3 log n)-approximation [Chuzhoy, Khanna 09]

2 -connectivity problems (like ours) 2 -edge-connected subgraph spanning k vertices: O(log n log 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(log 3 n) [Antonakopoulos, Chekuri, Shepherd, Zhang 07]

Our results Problem Edge case Vertex case FTGS-2 3. 55 VC hard O(log 2 Our results Problem Edge case Vertex case FTGS-2 3. 55 VC hard O(log 2 n) O(k log 2 n) O( n log n) GS hard (each group has 2 vertices) FTGS-k (each group has k vertices) FTGS (with disjoint groups) FTGS directed Label Cover Hard FTGS = Fault tolerant group Steiner, GS = Group Steiner, VC = Vertex cover

Our results Problem Edge case Vertex case FTGS-2 3. 55 VC hard O(log 2 Our results Problem Edge case Vertex case FTGS-2 3. 55 VC hard O(log 2 n) O(k log 2 n) O( n log n) GS hard (each group has 2 vertices) FTGS-k (each group has k vertices) FTGS (with disjoint groups) FTGS directed Label Cover Hard FTGS = Fault tolerant group Steiner, GS = Group Steiner, VC = Vertex cover

Why is our problem difficult? Known algorithms for Group Steiner tree are based on 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?

Algorithm for FTGS-2 (edge case) As |gi| = 2, all terminals must be connected 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.

Violated sets Say that X V is violated if there is only one edge 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”.

Why are violated sets uncrossable? For any violated set X, the set X ∩ 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. gi X∩T Subtrees are laminar! (i. e. , either two subtrees are disjoint or one is contained in the other. )

The two cases X-Y=X and Y-X=Y g 1 g 2 X∩T Y∩T X∩Y=Y and The two cases X-Y=X and Y-X=Y g 1 g 2 X∩T Y∩T X∩Y=Y and XUY=X Y∩T g 2 g 1 X∩T

Consequence The problem of finding a minimum cost cover of an uncrossable family admits 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.

Algorithm for FTGS-2 (vertex case) First step: Steiner tree (same) Second step: Augmentation problem Algorithm for FTGS-2 (vertex case) First step: Steiner tree (same) Second step: Augmentation problem is now different u u 1 g u 2

The augmentation problem Theorem: The group g is satisfied iff either u 1 or The augmentation problem Theorem: The group g is satisfied iff either u 1 or u 2 is 2 -vertex-connected with r. u u 1 g u 2

The augmentation problem r u u 1 g u 2 The augmentation problem r u u 1 g u 2

The augmentation problem Profit(v) = number of groups g for which v serves the The augmentation problem Profit(v) = number of groups g for which v serves the role of either u 1 or u 2 u u 1 g u 2

Density version of 2 -vertex-connected graph problem Given a graph with profits on vertices, 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(log 2 n)-approximation.

FTGS-k (|g| ≤ k for all groups g) A similar argument with a careful FTGS-k (|g| ≤ k for all groups g) A similar argument with a careful counting gives O(k log 2 n) approximation if groups are assumed to be disjoint.

Thanks! Thanks!

How many groups can u 1 or u 2 cover? r P=4 P=3 g How many groups can u 1 or u 2 cover? r P=4 P=3 g 2 P=1 P=2 g 3 g 1 P=1 g 3 g 4 g 2 g 3 g 4 g 1 g 3