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: with costs: where where

4 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. Motivation (cont’d) bandwidth cost Question: satisfy bandwidth for a set of demands by installing sufficient capacities at minimum cost

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 s i to t i. Goal: minimize the total cost

If this edge is bought its contribution to total cost is 14. If this edge is rented, its contribution to total cost is 2x3=6 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). as the per use cost (length).

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

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] [AA’97, Bartal’98, FRT’03] O(log n) approx for the single-sink case [MMP’00] 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(log 1/2-  n) in general [Andrews’04], unless NP  ZPTIME(n polylog(n) ) Hardness of Ω(log log n) for the single-sink case [CGNS’05] and Ω(log 1/2-  n) in general [Andrews’04], unless NP  ZPTIME(n polylog(n) ) Constant approx for several special cases: [AKR’91,GW’95,KM’00,KGR’02,KGPR’02,GKR’03] Recently we gave an O(log 4 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 .D where 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(log 3 n.opt). Corollary: we get an (O(log 2 n),O(log 3 n))-bicriteria approx for shallow-light k-Steiner tree Theorem 2: There is an O(log 4 n)-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: 1. Run the ( ,  )-approx alg A for (k/2)-Steiner tree with diameter bound D=4OPT/k 2. 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*. 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 G c 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: 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 G c then we can find a solution of cost X and diameter Y in G. So it is enough to work with G c. 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 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 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 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 approximation algorithm for buy-at-bulk k-steiner trees. The current lower bound is only Ω(log log n). We obtain O(log 4 n) 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. Thank you.