1. Approximation Algorithms for Buy-at-Bulk Network Design MohammadTaghi Hajiaghayi Labs- Research Labs- Research

2. 2 Motivation Suppose we are given a network and some nodes have to be connected by cables 10 12 8 21 27 11 5 9 14 7 21 3 16 Each cable has a cost (installation or cost of usage) Question: Install cables satisfying demands at minimum cost This is the well-studied Steiner forest problem and is NP-hard

3. 3 Motivation (cont’d) Consider buying bandwidth to meet demands between pairs of nodes. The cost of buying bandwidth satisfy economies of scale The capacity on a link can be purchased at discrete units: Costs will be: Costs will be: Where Where

4. 4 So if you buy at bulk you save More generally, we have a non-decreasing monotone concave (or more generally sub-additive) function where f (b) is the minimum cost of cables with bandwidth b. Motivation (cont’d) bandwidth cost  Question: Given a set of bandwidth demands between nodes, install sufficient capacities at minimum total cost The problem is called Multi-Commodity Buy-at-Bulk (MC-BB) Multi-Commodity Buy-at-Bulk (MC-BB)

5. 5 Motivation (cont’d) The previous problem is equivalent to the following problem: There are a set of pairs to be connected to be connected For each possible cable connection e we can: Buy it at b(e): and have unlimited use Buy it at b(e): and have unlimited use Rent it at r(e): and pay for each unit of flow 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

6. 6 Motivation (cont’d) 10 14 3 If this edge is bought its contribution to total cost is 14. If this edge is rented, its contribution to total cost is 2*3=6 Total cost is: where f(e) is the number of paths going over e.

7. 7 These problem is equivalent to the cost-distance problem: cost function cost function length function length function Also a set of pairs of nodes each with a demand for every i Feasible solution: a set s.t. all pairs are connected in are connected in Cost-Distance

8. 8 Cost-Distance (cont’d) The cost of the solution is: where is the shortest path in The cost is the start-up cost and is the per-use cost (length). is the per-use cost (length). Goal: minimize total cost.

9. 9 Multicommodity Buy At Bulk Note that the solution may have cycles The problem is called Multi-Commodity Buy-at-Bulk (MC-BB) Multi-Commodity Buy-at-Bulk (MC-BB) 5 11 8 21 12

10. 10 Special Cases If all s i (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 : Uniform-case 5 11 8 21 12 Single-source

11. 11 Previous Work Previous Work Formally introduced by [Salman, Cheriyan, Ravi, and Subramanian ’97] O(log n) approximation for the uniform case, i.e. each edge e has cost c+l  f(e) for some fixed constants c, l [Awerbuch and Azar ’97], [Bartal’98] O(log n) approximation for the uniform case, i.e. each edge e has cost c+l  f(e) for some fixed constants c, l [Awerbuch and Azar ’97], [Bartal’98] O(log n) randomized approximation for the single- sink case O(log n) randomized approximation for the single- sink case [Meyerson, Munagala, and Plotkin ’00] [Meyerson, Munagala, and Plotkin ’00] O(log n) deterministic approximation for the single-sink case [Chekuri, Khanna, and Naor ’01] [Chekuri, Khanna, and Naor ’01]

12. 12 Hardness Results for Buy-at-Bulk Problems Hardness of Ω(log log n) for the single- sink case [ Chuzhoy, Gupta, Naor, and Sinha ’05] Hardness of Ω(log log n) for the single- sink case [ Chuzhoy, Gupta, Naor, and Sinha ’05] Ω(log 1/2-  n) in general [Andrews ’04], unless NP  ZPTIME(n polylog(n) ) Ω(log 1/2-  n) in general [Andrews ’04], unless NP  ZPTIME(n polylog(n) )

13. 13 Algorithms for Special Cases Steiner Forest [Agrawal, Klein, and Ravi ’91] [Goemans and Williamson ’95] Single source [Guha, Meyerson, and Munagala ’01] [Talwar ’02] [Gupta, Kumar, and Roughgarden ’02] [Meyerson, Munagala, and Plotkin ’00] [Goel and Estrin ’03]

14. 14 Multicommodity Buy at Bulk Multicommodity Uniform Case: [Awerbuch and Azar ’97] [Awerbuch and Azar ’97] [Bartal ’98] [Bartal ’98] [Gupta, Kumar, Pal, and Roughgarden ’03] [Gupta, Kumar, Pal, and Roughgarden ’03] The only known approximation for the general case was [Charikar and Karagiozova’05]. The ratio was (D is the max demand) exp( Õ ( log 1/2 (nD) )) exp( Õ ( log 1/2 (nD) ))

15. 15 Our Main Result [ Chekuri, Hajiaghayi, Kortsarz, Salavatipour, FOCS’06] Theorem: If h is the number of pairs of s i,t i then there is a polytime algorithm with approximation ratio O(log 4 h). For simplicity we focus on the unit-demand case (i.e. d i =1 for all i’s) and we present Õ (log 5 n). Õ (log 5 n).

16. 16 Overview of the Algorithm The algorithm iteratively finds a partial solution connecting some of the residual pairs The new pairs are then removed from the set; repeat until all pairs are connected (routed) Density of a partial solution = cost of the partial solution cost of the partial solution # of new pairs routed # of new pairs routed The algorithm tries to find low density partial solution at each iteration

17. 17 Overview of the Algorithm (cont’d) The density of each partial solution is at most Õ(log 4 n)  (OPT / h') where OPT is the cost of optimum solution and h' is the number of unrouted pairs Õ(log 4 n)  (OPT / h') where OPT is the cost of optimum solution and h' is the number of unrouted pairs A simple analysis (like for set cover) shows: Total Cost Total Cost  Õ(log 4 n)  OPT  (1/n 2 + 1/(n 2 - 1) +…+ 1)  Õ(log 4 n)  OPT  (1/n 2 + 1/(n 2 - 1) +…+ 1)  Õ(log 5 n)  OPT  Õ(log 5 n)  OPT

18. 18 Structure of the Optimum How to compute a low-density partial solution? Prove the existence of low-density one with a very specific structure: junction-tree Junction-tree: given a set P of pairs, tree T rooted at r is a junction tree if It contains all pairs of P It contains all pairs of P For every pair s i,t i  P the For every pair s i,t i  P the path connecting them path connecting them in T goes through r in T goes through r r

19. 19 Structure of the Optimum (cont’d) So the pairs in a junction tree connect via the root We show there is always a partial solution with low density that is a junction tree Observation: If we know the pairs participating in a junction-tree it reduces to the single-source BB problem r Then we could use the O(log n) approximation of [MMP’00]

20. 20 Summary of the Algorithm So there are two main ingredients in the proof Theorem 2: There is always a partial solution that is a junction tree with density Õ (log 2 n)  (OPT / h') Theorem 3: There is an O (log 2 n) approximation for the problem of finding lowest density junction tree (this is low density SS-BB). Corollary: We can find a partial solution with density Õ (log 4 n)  (OPT / h') This implies an approximation Õ (log 5 n) for MC-BB. This implies an approximation Õ (log 5 n) for MC-BB.

21. 21 More Details of the Proof of Theorem 2: We want to show there is a partial solution that is a junction tree with density Õ (log 2 n)  (OPT / h') Consider an optimum solution OPT. Let E* be the edge set of OPT, OPT c be its cost and OPT l be its length.

22. 22 More Details of the Proof of Theorem 2: By the result of [Elkin, Emek, Spielman, and Tang ’05] on probabilistic distribution on spanning trees and by loosing a factor Õ (log 2 n) on length, we can assume that E* is a forest T (WLOG we assume T is connected). 5 11 8 21 12  Note that OPT may have cycles

23. 23 More Details of the Proof of Theorem 2: From T we obtain a collection of rooted subtrees T 1,…,T a such that any edge e of T is in at most O(log n) of the subtrees For every pair there is exactly one index i such that both vertices are in T i ; further the root of T i is their least common ancestor The total cost of the junction trees is at most Õ (log 2 n)  OPT (O (log n)  OPT c + Õ (log 2 n)  OPT l ) Õ (log 2 n)  OPT (O (log n)  OPT c + Õ (log 2 n)  OPT l ) Thus at least one of junction trees of T 1,…,T a has the desired density of Õ (log 2 n)  (OPT / h')

24. 24 More Details of the Proof of Theorem 2: Given T, we pick a centeroid r 1 (i.e., largest remaining component has at most 2/3 |V(T)| vertices). Add tree T rooted at r 1 to the collection Remove r 1 from T and apply the procedure recursively to each of the resulting component Each pair is on exactly one subtree in the collection in the collection Depth of recursion is in O (log n) r

25. 25 Some Details of the Proof of Theorem 3: Theorem 3: There is an O(log 2 n) approximation for finding lowest density junction tree. This is very similar to SS-BB except that we have to find a lowest density solution. Here we have to connect a subset of the pairs to the root r with lowest density the pairs to the root r with lowest density (= cost of solution / # of pairs in sol). (= cost of solution / # of pairs in sol). Let denote the set of paths from r to i. Let denote the set of paths from r to i. We formulate the problem as an IP and then consider the LP relaxation of the problem r

26. 26 Some Details of the Proof of Theorem 3: We solve the LP by setting y s =y t for each pair (s,t), and then find a subset of nodes to solve the SS-BB We find a class of y among O (log n) classes of almost equal y i with maximum sum and scale up(loose a factor O (log n)) We use the O (log n) approx of [MMP’00,CKN’01] for SS- BB (indeed it is upper bound on integrality gap of the LP)

27. 27 Some Remarks: For the polynomially bounded demand case we can find low density junction-trees using a more refined region growing technique and also using a greedy algorithm (within O (log 4 n)) [Hajiaghayi, Kortsarz and Salavatipour, ECCC’ 06] [Hajiaghayi, Kortsarz and Salavatipour, ECCC’ 06] The greedy algorithm is based on an algorithm for the k-shallow-light tree problem [Hajiaghayi, Kortsarz, and Salavatipour, APPROX ’06] k-shallow-light tree problem [Hajiaghayi, Kortsarz, and Salavatipour, APPROX ’06] There is a conjectured upper bound of O (log n) for distortion of embedding a graph metric into a probability distribution over its spanning tree [Alon, Karp, Peleg, and West ’91] If true, that would improve our approximation factor for arbitrary demands to O (log 4 n)

28. 28 Some Remarks (cont’d): Indeed, as suggested by Racke, our current approach can be applied via Bartal’s trees (and interestingly not FRT) to obtain an O(log h) factor instead of Õ (log 2 h) factor For a constant fraction of the pairs, we use strong diameter property which is true in Bartal’s construction It is more technical, but we can obtain factor O (log 4 h) for general demands (solving an open problem)

29. 29 Natural Generalization: Group Cost-Distance Each edge has a buying and a renting cost. Subsets called demand groups. Each group only pays one rental cost on each edge

30. 30 Group Cost-Distance A solution connects each group S i using a tree T i F = union of edges in the trees T i f e : number of trees T i using edge e

31. 31 Group Cost-Distance By generalizing our current approach, we can obtain an O(log 6 n) approximation for this problem [Gupta, Hajiaghayi, and Kumar ’07]. [Gupta, Hajiaghayi, and Kumar ’07].. It is 2^O(log 1-e n) hard if each edge has different [Gupta, Hajiaghayi, and Kumar ’07]. grouping [Gupta, Hajiaghayi, and Kumar ’07].

32. 32 Recent Extensions The result O(log 4 n) can be extended to the vertex-weighted case but requires some new ideas and some extra work [CHKS’07]. Especially we obtain the tight result O(log n) for the single-sink vertex-weighted case via LP rounding Also our results can be extended to stochastic Steiner tree with non-uniform inflation (by loosing an extra factor O(log n)) [Gupta, Hajiaghayi, and Kumar ’07]. Some technique has been used in the Dial-a-Ride problem [Gupta, Hajiaghayi, Ravi, and Nagarajan ’07]. [Gupta, Hajiaghayi, Ravi, and Nagarajan ’07]. O(log 3 n) approximation for non-uniform buy at bulk when demands are polynomial [Kortsarz and Nutov’ 07] O(log 4 n) approximation when want to have two disjoint paths between each demand pair [Chekuri, Antonakapoulos, Shepherd and Zhang’ 07] O(n 1/2 ) approximation for generalized directed Steiner tree [Chekuri, Even, Gupta, and Segev’ 08]. Oblivious network design with ratio O(log 3 n) for uniform buy at bulk, i.e., costs of all edges are the same sub-additive function f [Gupta, Hajiaghayi, and Raecke ’07].

33. 33 Open Problems There are still quite large gaps between upper bounds (approx alg) and lower bounds (hardness) For MC-BB: vs For MC-BB: vs For SS-BB: vs For SS-BB: vs It would be nice to upper bound the integrality gap for MC-BB. It would be nice to upper bound the integrality gap for MC-BB. Emphasize on the conjecture of [Alon, Karp, Peleg, and West’ 91]

34. 34 Thanks for your attention… تشکر Obrigado