1. Improved Approximation Algorithms for Directed Steiner Forest Moran Feldman Technion Joint work with: Guy Kortsarz,Rutgers University Camden Zeev Nutov,The Open University of Israel

2. Talk Outline 2 Presenting the problems Prior art and our results Previous algorithm for Directed Steiner Forest (DSF) Our algorithms for k-DSF and DSFOur algorithms for k-DSF and DSF Summary

3. Problem 1: Directed Steiner Forest (DSF) 3 Instance: A digraph G = (V,E) with edge costs c(e) and a set D  S  T of ordered node pairs of V. Objective: Find a subgraph H  G of minimum cost containing an s-t path for every (s,t)  D. Problem 2: k-Directed Steiner Forest (k-DSF) Instance: As for DSF and an integer 0 ≤ k ≤ |D|. Objective: Find a subgraph H  G of minimum cost containing an s-t path for (at least) k pairs (s,t)  D.

4. ProblemUndirectedDirected In terms of n In terms of k In terms of n In terms of k Steiner Forest 2 [AKR 95] O(n 1+ε )O(k 1/2+ε ) [CEGS 08] k-Steiner Forest O(n 1/2 ) [GHNR 07] O(k 1/2 ) [GHNR 07] O(n 4/3 ) [CCCDG 99] O(k 2/3 ) [CCCDG 99] ProblemUndirectedDirected In terms of n In terms of k In terms of n In terms of k Steiner Forest 2 [AKR 95] O(n 1+ε )O(k 1/2+ε ) * [CEGS 08] k-Steiner Forest O(n 1/2 ) [GHNR 07] O(k 1/2 ) [GHNR 07] O(n 1+ε ) O(n 4/3 ) O(k 1/2+ε ) O(k 2/3 ) ProblemUndirectedDirected In terms of n In terms of k In terms of n In terms of k Steiner Forest 2 [AKR 95] O(n 4/5+ε ) O(n 1+ε ) O(k 1/2+ε ) * [CEGS 08] k-Steiner Forest O(n 1/2 ) [GHNR 07] O(k 1/2 ) [GHNR 07] O(n 1+ε ) O(n 4/3 ) O(k 1/2+ε ) O(k 2/3 ) Prior Art 4 Theorem 2 DSF admits an O(n 4/5+ε ) approximation scheme. Theorem 1 k-DSF admits a combinatorial O(k 1/2+ε ) approximation scheme. Note that k-SF and k-DSF have (almost) the same ratio in terms of k. and Our Results First sublinear algorithm in terms of n. Lower bound Ω(n 0.5 ) [DK 99]. A variant of the algorithm gives an O(m 2/3+ ε ) approximation.

5. 5 The Density Problem Instance: As in k-DSF. Objective: Find a subgraph H  G of minimum density: min{k, # of pairs connected by H} c(H)c(H) Reductions to Density Using set-cover type analysis, both reductions preserve the approximation ratio (up to a logarithmic factor, for low ratios). Approximation for density with k = |D| Approximation for DSF Approximation for k-DSF Approximation for density

6. Junction Trees 6 Definition A junction-tree is a union of: An in-going tree rooted at r An out-going tree rooted at r r It is easy to see that in every graph there is a junction tree of density: k 1/2 ∙ opt/k [Chekuri, Even, Gupta, Segev SODA08]

7. Algorithm for DSF of [CEGS 08] 7 Difficulty Finding the best junction tree is NPC, it must be approximated. Solution Reduction to Group Steiner Forest, and approximating it via an LP-relaxation. Yields an approximation ratio of O(k 1/2+ε ). The LP is suitable only for DSF (the case k = |D|). What’s Now? What do we know about junction trees? There is a good density junction tree in every instance of k-DSF.  Nobody knows how to find such a junction tree in the general settings of k-DSF. Idea Use junction trees to approximate the Density Problem.

8. Theorem 1 k-DSF admits an O(k 1/2+ε ) approximation scheme. Junction Star-Tree: A union of disjoint: In-going star from S to r Out-going tree from r to T r

9. Algorithm for k-DSF 9 Simple Reductions Ideas Use junction star-trees to approximate the Density Problem. Finding good density junction star-tree via a reduction to: Instance: A digraph G = (V, E) with edge costs, a root node r, a set U  V of terminals, and an integer k. Objective: Find a min-cost subtree T of G rooted at r spanning k terminals. Approximation: We need the density version: “Find the best density tree”. It has an O(k ε ) approximation scheme by [CCCDG 99]. AssumptionJustification Metric costs.Metric completion. No edge enters a node of S or leaves a node of T. Create a new source and terminal for every node v, connect them to v by zero cost edges. A node belongs to at most one pair of D.Multiply nodes belonging to multiple pairs. k-Directed Steiner Tree (k-DST)

10. Finding a good density junction star-tree 10 A junction star-tree:A tree rooted at r: t1t1 t2t2 t3t3 t4t4 t5t5 t' 1 t' 2 t' 3 t' 4 t' 5 r s1s1 s2s2 s3s3 s4s4 s5s5 t1t1 t2t2 t3t3 t4t4 t5t5 r Reduction to k-DST We guess r No other information about the junction star-tree is needed

11. Concluding the Algorithm 11 What do we get? Using junction star-trees to approximate density, we get an O(k 1/2+ε )-approximation algorithm for k-DSF. Comparison with the algorithm of [CEGS 08] Finding a good augmentation: Easy, works for k-DSF as well. Proving existence of a good augmentation: Non-trivial. Advantage: Combinatorial algorithm, does not solve LPs. Theorem In every instance of k-DSF (after metric completion) there is a junction star-tree of density at most: (8k) 1/2 ∙ opt/k. Long proof, see the paper for the details. Based on averaging arguments.

12. Theorem 2 DSF admits an O(n 4/5+ε ) approximation scheme.

13. Algorithm Overview 13 Notation A path is short if its length ≤ opt/n 4/5, otherwise it is long (opt is known). U(s,t) – The set of nodes having both short path from s, and short path to t. A pair (s, t)  D is good if |U(s,t)| ≥ n 2/5, and bad otherwise. s v1v1 v2v2 vtvt t U(s,t)U(s,t) Short Paths Few nodes  Bad Pair Many nodes  Good Pair

14. Algorithm Overview - Continue 14 Connecting Good Pairs Put every node v into a set R with probability p = 2ln k / n 2/5. Connect every good pair with short path via R, if possible. Connecting Bad Pairs Let L  D be the set of pairs connected by long paths in OPT.  Case 1: |L| ≥ ½|D|: Good density junction tree.  Case 2: |L| < ½|D|: Good density via LP rounding. Approximate the density problem by the better density edge set. A pair (s, t) is connected if R  U(s,t)   Pr[R  U(s,t) =  ] ≤ 1/k 2 for good pairs By the union bound, all good pairs are connected with probability ≥ 1-1/k By the Chernoff bound, with probability 1-1/k, the cost is no more than: Henceforth, we assume that D contains only bad pairs.

15. Case 1: |L| ≥ ½|D| (reduction to [CEGS 08]) 15 By an averaging argument there is a junction tree of of density: The method of [CEGS 08], let us find a junction tree of density: Case 2: |L| < ½|D| (LP-rounding) This LP asks to connect at least half of the bad pairs using short paths:  (i) – The set of short paths connecting the i-th bad pair.  – The set of short paths connecting any bad pair. (An average flow of ½ or more) (The flow of a pair is the sum of the flow paths) (Pay for edges along the flow path)

16. 16 Using the LP At least |D|/3 pairs have flow  ¼, consider a cut in the graph separating such pair (s, t): s1s1 s2s2 spsp t1t1 t2t2 tqtq ts (s,t) bad pair  p+q < n 2/5 The number of edges (carrying flow) crossing the cut < p∙q < n 4/5 /4. The flow  ¼, some edge crossing the cut has flow  1/n 4/5. Difficulty: The number of variables might be exponential. Solution: Approximate separation oracle for the dual program. Derives a solution of cost opt∙(1+ε) for the LP in polynomial time. Round the x e variables: F = {e | x e ≥ 1/n 4/5 } Cost: c(F) = O(n 4/5 ) ∙ opt # of pair connects: at least |D|/3 bad pairs Density: O(n 4/5 )∙opt/|D|

17. Open Questions 17 Regarding our algorithm for DSF:  Can its ratio be improved?  Can it be extended to k-DSF as well?  A ratio better than O(n 1/2 ) is unlikely, due to a reduction to LABEL-COVER max [DK 99]. Generalizations:  DSF and k-DSF have a generalization where every pair (s,t)  D has a demand r(s,t) and it must be connected by r(s,t) edge disjoint paths.  Can any of the results presented be extended to this generalization of the corresponding problem?  No results are known for this generalization, even when the demands are limited to 2.