Tree Embeddings for 2-Edge-Connected Network Design Problems

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

2. Running Example: Group Steiner Problems Given: graph G = (V,E), edge costs c: E → R, root vertex r. Set of groups X1, X2, …, Xk with each Xi⊆ V. Goal: output subgraph H ⊆ G such that all groups are connected to the root r. (Xi is connected to r if some vi∈Xi is connected to r in H) Cost:c(H) = e ∈ H c(e) Group Steiner Tree [Garg et al SODA 1998] O(log3 n)-approximation algorithm

3. An Illustration root r group X1 group X3 group X2 group X4 Big advantage: optimal solution is always a tree!

4. 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. root r group X1 group X3 group X2 group X4

5. 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 Xi is 2-edge-connected to r if  e, some vi∈Xi is connected to r in H \ e Cost:c(H) = e ∈ H c(e) 2-Edge-Connected Group Steiner (2-ECGS Problem)

6. Known Results Khandekar et al. [FSTTCS 2009] show the following results: Our Result O(log4 n)-approximation algorithm for 2-ECGS

7. Other Results

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

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

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

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

13. 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 xe/xp(e) • If not, don’t include e GKR SODA 1998 o.5 o.2 o.2 o.4 o.2 o.2 o.1 o.1

14. Rounding Continued.. After Stage I • Expected cost incurred by each edge e is c(e) xe • Each group is connected to root with reasonable probability. Stage II: Non-Tree Rounding • Consider any non-tree edge f • Let e1 and e2 be the lowest edges “chosen” in stage I (on the cycle Of) • Changeyf to yf/xe1 + yf/xe2 e2 e1 f

15. 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

16. 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 r V3 ∈X3 V1 ∈X1 V2 ∈X2

17. Expected Cost • Stage I: O(1) c(OPT) • Stage II: O(log n) c(OPT) e1 Expected value of “scaled yf” = Pr[e1 is lowest edge] yf/xe1 + Pr[e2 is lowest edge] yf/xe2 + … ≤ xe1 (yf/xe1) + xe2 (yf/xe2) + … ≤ O(log n) yf e2 e3 f Only distinct powers of ½ matter! Assume xe’s are powers of ½

18. Summary • Showed O(log4 n)-approximation for 2-ECGS • Similar techniques also work for • Open Questions • Better approximation for 2-ECGS (lower bound is Ω(log2 n)) • k-edge-connectivity for larger values of k?

19. Thank You! Questions?