1 Tree Embeddings for 2-Edge-Connected Network Design Problems Ravishankar Krishnaswamy Carnegie Mellon University joint work with Anupam Gupta and R. Ravi

Running Example: Group Steiner Problems Given: graph G = (V,E), edge costs c: E → R, root vertex r. Set of groups X 1, X 2, …, X k with each X i ⊆ V. Goal: output subgraph H ⊆ G such that all groups are connected to the root r. (X i is connected to r if some v i ∈ X i is connected to r in H) Cost: c(H) =   e ∈ H c(e) 2 Group Steiner Tree [Garg et al SODA 1998] O(log 3 n)-approximation algorithm Group Steiner Tree [Garg et al SODA 1998] O(log 3 n)-approximation algorithm

An Illustration 3 root r group X 1 group X 2 group X 3 group X 4 Big advantage: optimal solution is always a tree!

What if we require fault-tolerance? Solution must be robust to “failure” of any single edge. Need to reinforce with back-up paths, to eliminate all such cut-edges. 4 root r group X 1 group X 2 group X 3 group X 4

What if we require fault-tolerance? Solution must be robust to “failure” of any single edge. Goal: output subgraph H ⊆ G such that all groups are 2-edge-connected to the root r. A Group X i is 2-edge-connected to r if  e, some v i ∈ X i is connected to r in H \ e Cost: c(H) =   e ∈ H c(e) 5 2-Edge-Connected Group Steiner (2-ECGS Problem) 2-Edge-Connected Group Steiner (2-ECGS Problem)

6 Known Results Khandekar et al. [FSTTCS 2009] show the following results: ProblemApproximability 2-ECGS (group size 2) VCGS (group size 2)O(log 2 n) 2-ECGS (group size q)O(q log 2 n) 2-VCGS (group size q)O(q log 2 n) Our Result O(log 4 n)-approximation algorithm for 2-ECGS

Other Results 7 ProblemApproximation Factor 2-EC Facility LocationO(log n) 2-EC Buy-at-BulkO(log 2 n) [ACSZ FOCS07] O(log 3 n) for 2-VC-BaB with 1 cable type 2-EC k-SubgraphO(log 3 n) [LNSS J. Comp 2009] O(log 2 n)

8 A Structure Property Consider any group X i Any solution must resemble the following two types group X i r r type Assumption only for the talk! Assumption only for the talk!

9 Our High-level Approach Embed graph into random subtree [Abraham et al. FOCS 2008] – get better structure on edge costs [GKR STOC 2009] Solve first stage problem of 1-connecting the groups – using existing LP based algorithm [Garg et al. SODA 1998] Solve the augmentation problem to get 2-connectivity – show that there exists a low-cost augmentation – this is where subtree embedding comes in handy

10 Step 1: Backboned Graphs 1. Find a random low-stretch spanning subtree T (the base tree) [ABN08] 2. Set cost of any non-tree edge to be the cost of the base-tree path. r ℓ a b c d ℓ = a + b + c + d x y There are at most m fundamental cycles Cost is comparable to non-tree edge Fundamental Cycle E[ ℓ ] ≤ O(log n) c(x,y)

11 2-ECGS on Backboned Graphs r vivi Consider a backboned graph with base tree T (the red edges) Consider some group X i Let OPT 2-edge-connect some v i ∈ X i to the root r Without loss of generality OPT buys the r-v i base tree path. Consider a cut-edge on this path. Look at the cut this induces on the base tree. Some edge of OPT must cross this cut. Get a covering cycle of twice the cost! 1.Every group has a tree path from r to some vertex v i in OPT 2.Each edge on this tree path has a “covering cycle” in OPT

2-ECGS LP Formulation (on Backboned Graphs) 12 x e -- tree edge e is included in the solution y f -- non-tree edge f is included in the solution x e -- tree edge e is included in the solution y f -- non-tree edge f is included in the solution 1.Every group has a tree path from r to some vertex v i in OPT 2.Each edge on this tree path has a “covering cycle” in OPT

The Rounding Strategy Stage I: Tree Rounding From the root, traverse the tree top down For an edge e, check if parent edge p(e) has been included – If so, include e in the solution with probability x e /x p(e) – If not, don’t include e 13 GKR SODA 1998 o.5 o.2 o.1 o.2 o.4 o.2 o.1 o.2

Rounding Continued.. After Stage I Expected cost incurred by each edge e is c(e) x e Each group is connected to root with reasonable probability. 14 Stage II: Non-Tree Rounding Consider any non-tree edge f Let e 1 and e 2 be the lowest edges “chosen” in stage I (on the cycle O f ) Change y f to y f /x e 1 + y f /x e 2 f e1e1 e2e2

Rounding Continued.. The scaled solution is feasible to the “augmentation LP” This LP solution is a fractional set-cover! – Can be rounded using several techniques 15

Putting the Pieces Together After Stage I - Singly-connect each group with reasonable probability After Stage II - Cover every chosen tree edge by some cycle All groups connected in Stage I are now 2-edge-connected 16 r V 1 ∈ X 1 V 2 ∈ X 2 V 3 ∈ X 3

Expected Cost Stage I: O(1) c(OPT) Stage II: 17 f e1e1 e2e2 e3e3 Expected value of “scaled y f ” = Pr[e 1 is lowest edge] y f /x e 1 + Pr[e 2 is lowest edge] y f /x e 2 + … ≤ x e1 (y f /x e 1 ) + x e 2 (y f /x e 2 ) + … ≤ O(log n) y f Expected value of “scaled y f ” = Pr[e 1 is lowest edge] y f /x e 1 + Pr[e 2 is lowest edge] y f /x e 2 + … ≤ x e1 (y f /x e 1 ) + x e 2 (y f /x e 2 ) + … ≤ O(log n) y f O(log n) c(OPT) Only distinct powers of ½ matter!Assume x e ’s are powers of ½

18 Summary Showed O(log 4 n)-approximation for 2-ECGS Similar techniques also work for Open Questions – Better approximation for 2-ECGS (lower bound is Ω(log 2 n)) – k-edge-connectivity for larger values of k? ProblemApproximation Factor 2-EC Facility LocationO(log n) 2-EC Buy-at-BulkO(log 2 n) [ACSZ FOCS07] O(log 3 n) for 2-VC-BaB with 1 cable type 2-EC k-SubgraphO(log 3 n) [LNSS J. Comp 2009] O(log 2 n)

19 Thank You! Questions?