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Improved Approximation Algorithms for Directed Steiner Forest

Explore improved techniques for solving Directed Steiner Forest problems, with a focus on k-DSF and density, featuring novel approximation schemes and junction trees. See how junction star-tree structures enhance algorithm efficiency.

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Improved Approximation Algorithms for Directed Steiner Forest

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  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 • Presenting the problems • Prior art and our results • Previous algorithm for Directed Steiner Forest (DSF) • Our algorithms for k-DSF and DSF • Summary

  3. Problem 1: Directed Steiner Forest (DSF) 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. and Our Results Prior Art Theorem 2 DSF admits an O(n4/5+ε) approximation scheme. Theorem 1 k-DSF admits a combinatorial O(k1/2+ε) approximation scheme. Note that k-SF and k-DSF have (almost) the same ratio in terms of k. First sublinear algorithm in terms of n. Lower bound Ω(n0.5) [DK 99]. A variant of the algorithm gives an O(m2/3+ ε) approximation.

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

  6. r Junction Trees • Definition • A junction-tree is a union of: • An in-going tree rooted at r • An out-going tree rooted at r It is easy to see that in every graph there is a junction tree of density: k1/2 ∙ opt/k [Chekuri, Even, Gupta, Segev SODA08]

  7. Algorithm for DSF of [CEGS 08] Idea Use junction trees to approximate the Density Problem. 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(k1/2+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.

  8. r Proof of Theorem 1 Theorem 1 k-DSF admits an O(k1/2+ε) approximation scheme. Junction Star-Tree: A union of disjoint: In-going star from S to r Out-going tree from r to T

  9. Algorithm for k-DSF • Simple Reductions • Ideas • Use junction star-trees to approximate the Density Problem. • Finding good density junction star-tree via a reduction to: k-Directed Steiner Tree (k-DST) 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(kε) approximation scheme by [CCCDG 99].

  10. A tree rooted at r: s4 s5 s3 s2 s1 r r t1 t2 t3 t4 t5 t1 t2 t3 t4 t5 t'1 t'2 t'3 t'4 t'5 Finding a good density junction star-tree A junction star-tree: • Reduction to k-DST • We guess r • No other information about the junction star-tree is needed

  11. Concluding the Algorithm 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. • What do we get? • Using junction star-trees to approximate density, we get an O(k1/2+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.

  12. Proof of Theorem 2 Theorem 2 DSF admits an O(n4/5+ε) approximation scheme.

  13. Short Paths Short Paths v1 v2 vt s t U(s,t) Many nodes  Good Pair Few nodes  Bad Pair Algorithm Overview • Notation • A path is short if its length ≤ opt/n4/5, 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)| ≥ n2/5, and bad otherwise.

  14. A pair (s, t) is connected if R  U(s,t)   • Pr[R  U(s,t) = ] ≤ 1/k2 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: Algorithm Overview - Continue Connecting Good Pairs • Put every node v into a set R with probability p = 2ln k / n2/5. • Connect every good pair with short path via R, if possible. Henceforth, we assume that D contains only bad pairs. 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.

  15. Case 1: |L| ≥ ½|D| (reduction to [CEGS 08]) • 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: (An average flow of ½ or more) (Pay for edges along the flow path) (The flow of a pair is the sum of the flow paths) (i) – The set of short paths connecting the i-th bad pair.  – The set of short paths connecting any bad pair.

  16. s1 t1 s2 t2 s t sp tq Using the LP • 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 xe variables: F = {e | xe ≥ 1/n4/5} • Cost: c(F) = O(n4/5) ∙ opt • # of pair connects: at least |D|/3 bad pairs • Density: O(n4/5)∙opt/|D| At least |D|/3 pairs have flow  ¼, consider a cut in the graph separating such pair (s, t): • (s,t) bad pair p+q < n2/5 • The number of edges (carrying flow) crossing the cut < p∙q < n4/5/4. • The flow  ¼, some edge crossing the cut has flow  1/n4/5.

  17. Open Questions • Regarding our algorithm for DSF: • Can its ratio be improved? • Can it be extended to k-DSF as well? • A ratio better than O(n1/2) is unlikely, due to a reduction to LABEL-COVERmax [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.

  18. Questions?

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