Approximating Buy-at-Bulk and Shallow-Light k -Steiner Trees

1. Approximating Buy-at-Bulk and Shallow-Light k-Steiner Trees Mohammad T. Hajiaghayi (CMU) Guy Kortsarz (Rutgers) Mohammad R. Salavatipour (U. Alberta) Presented by: Zeev Nutov

2. Definition of Buy-at-Bulk k-Steiner Tree • Given an undirected graph G(V,E), terminal set T V, a root sT, and integer k|T|. • Given two cost functions on the edges: • Buy cost • Rent cost • Goal: find a subtree H spanning at least k terminals including root s minimizing where

3. Motivation • Network design problems with two cost functions have many applications, e.g. in bandwidth reservation when we have economies of scale • Example: capacity on a link can be purchased at discrete units: with costs: where

4. Motivation (cont’d) • So if you buy at bulk you save • More generally, we have a concave function where f(b) is the minimum cost of cables with bandwidth b. Question: satisfy bandwidth for a set of demands by installing sufficient capacities at minimum cost cost bandwidth

5. Equivalent Cost Measure • Equivalent model: cost distance • There are a set of pairs to be connected • For each possible cable connection e we can: • Buy it at b(e):and have unlimited bandwidth • Rent it at r(e):and pay for each unit of flow • A feasible solution: buy and/or rent some edges to connect every sito ti. • Goal: minimize the total cost

6. If this edge is bought its contribution to total cost is 14. 10 14 If this edge is rented, its contribution to total cost is 2x3=6 3 Total cost is: where f(e) is the number of paths going through e.

7. Equivalent Cost Measure (cont’d) • If E’ is the set of edges of the solution, the cost is: where is the shortest path in • We can think of as the start-up cost and as the per use cost (length).

8. Special Cases • If all si’s (sources) are equal we have the single-source case (SS-BB) Single-source • If the cost and length functions on the edges are all the same, i.e. each edge e has costc+l×f(e) for constants c, l, we have the uniform case. 5 12 8 21 11

9. Known Results for Buy-at-Bulk Problems • Formally introduced by Salman et al. [SCRS’97] • O(log n)approximation for the uniform case [AA’97, Bartal’98, FRT’03] • O(log n)approx for the single-sink case [MMP’00] • Hardness of Ω(log log n) for the single-sink case [CGNS’05] and Ω(log1/2- n) in general [Andrews’04], unless NP ZPTIME(npolylog(n)) • Constant approx for several special cases: [AKR’91,GW’95,KM’00,KGR’02,KGPR’02,GKR’03] • Recently we gave an O(log4 n) approximation for the multicommodity case [HKS’06, CHKS’06] .

10. Shallow-Light k-Steiner Trees • Instances are similar to BB k-Steiner tree: • an undirected graph G(V,E), • terminals T V, • cost function, • length function, • a bound D and a parameter k  |T| • Find a tree spanning k terminals with minimum b-cost whose diameter under r-cost is at most D(assuming such a tree exists) • (,)-bicriteria approx: cost at most .opt and diameter is at most .Dwhere opt is the cost of optimum solution with diameter bound D

11. Our Results: Theorem 1: Given an instance of shallow-light k-Steiner tree with bound D, we find a (k/8)-Steiner tree with diameter O(log n.D) and cost O(log3n.opt). Corollary: we get an (O(log2n),O(log3n))-bicriteria approx for shallow-light k-Steiner tree Theorem 2: There is an O(log4n)-approximation for buy-at-bulk k-Steiner tree. Note: • BB k-Steiner generalizes k-MST and k-Steiner (when r=0). • Shallow-light k-Steiner generalizes shallow-light Steiner (when k=|T|) and k-MST (when D=1).

12. How to Reduce BB to Shallow-Light Let G be an instance of BB and assume we know the value of OPT (e.g. by guessing). Lemma: If there is an (,)-bicriteria algorithm A for shallow-light k-Steiner that finds a (k/8)-Steiner tree, then there is an O((+ ) log n) approx for BB k-Steiner. Proof: First, we can ignore every vertex with r-distance >OPT from the root. Then we run the following algorithm.

13. How to Reduce BB to Shallow-Light (cont’d) While k>0 repeat the following: • Run the (,)-approx alg A for (k/2)-Steiner tree with diameter bound D=4OPT/k • Decrease k by the number of terminals covered in the new solution; mark all these terminals as Steiner nodes; goto 1 The union of the solutions found is returned. Consider some iteration and let k’ be the number of unspanned terminals and H* be an optimal solution for BB k’-Steiner.

14. How to Reduce BB to Shallow-Light (cont’d) • Iteratively remove leaves (terminals) with r-distance > 2OPT/k’ from H*. • We delete at most k’/2 terminals and r-diameter is at most 4.OPT/k’ • Using alg A we find a (k’/16)-Steiner tree with diameter bound 4.OPT/k’. This adds at most k’..2OPT/k’=2.OPT to the rent cost; buy cost is at most .OPT • So we have covered a constant fraction of k’ at cost at most O((+).OPT). • A standard set-cover analysis shows the total cost is in O((+).OPT.log n).

15. Overview of Algorithm for Shallow-Light k-Steiner • First we compute a completion graph Gc of G : for every pair u,vV, compute (approximately) the minimum b-cost u,v-path with r-cost at most 2D. It is easy to show: Lemma: if there is a bicriteria solution of cost X and diameter Y in Gc then we can find a solution of cost X and diameter Y in G. • So it is enough to work with Gc. • Also, we can easily transform the un-rooted case and the rooted case to each other.

16. Overview of Algorithm … (cont’d) • We maintain a collection of trees • At the beginning every terminal is a tree of one node • We design a test that can fail or succeed • If the test succeds two trees are merged • Else some terminals are temporarily deleted

17. Overview of Algorithm … (cont’d) We maintain a collection of trees partition According to their number of terminals” 1 to 2 terminals 3 to 4 terminals p to 2p terminals

18. The Test • Pick a cluster of p to 2p terminals that contains ``many” roots • Every root is a terminal • A terminals is a TRUE terminal if belongs to the optimum • The test: does the collection of roots contain many terminals?

19. The Main Argument • If the test succeeds then two trees are contracted together at a low price • If it fails all roots in the cluster are removed • We loose “many” terminals • But only “few” true terminals • Hence eventually a tree will reach size k/8

20. Conclusion and Open Problems • We obtain O(log4n) approximation algorithm for buy-at-bulk k-steiner trees. The current lower bound is only Ω(log log n). • Main open problem: Can we improve the upper bound significantly or at least the lower bound to Ω(log n)?

21. Thank you.