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Approximation Algorithms for Buy-at-Bulk Network Design

Approximation Algorithms for Buy-at-Bulk Network Design. MohammadTaghi Hajiaghayi Labs- Research. Motivation. Suppose we are given a network and some nodes have to be connected by cables. Each cable has a cost (installation or cost of usage) Question : Install cables

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Approximation Algorithms for Buy-at-Bulk Network Design

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

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

  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: Where

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

  5. Motivation (cont’d) • The previous problem is equivalent to the following problem: • 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 use • 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. Motivation (cont’d) 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 2*3=6 3 Total cost is: where f(e) is the number of paths going over e.

  7. Cost-Distance • These problem is equivalent to the cost-distance problem: • cost 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

  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). • Goal: minimize total cost.

  9. Special Cases • If all si (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 cost • c + l f(e) for constants • c,l: Uniform-case 5 12 8 21 11

  10. 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)randomized approximation for the single-sink case [Meyerson, Munagala, and Plotkin ’00] • O(log n) deterministic approximation for the single-sink case [Chekuri, Khanna, and Naor ’01]

  11. Hardness Results for Buy-at-Bulk Problems • Hardness of Ω(log log n) for the single- sink case [Chuzhoy, Gupta, Naor, and Sinha ’05] • Ω(log1/2- n) in general [Andrews ’04], unless NP ZPTIME(npolylog(n))

  12. 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]

  13. Multicommodity Buy at Bulk Multicommodity Uniform Case: • [Awerbuch and Azar ’97] • [Bartal ’98] • [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)))

  14. Our Main Result [Chekuri, Hajiaghayi, Kortsarz, Salavatipour, FOCS’06] • Theorem: If h is the number of pairs of si,tithen there is a polytime algorithm with approximation ratio O(log4 h). • For simplicity we focus on the unit-demand case (i.e. di=1 for all i’s) and we present Õ(log5n).

  15. 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 # of new pairs routed • The algorithm tries to find low density partial solution at each iteration

  16. Overview of the Algorithm (cont’d) • The density of each partial solution is at most Õ(log4n)  (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  Õ(log4 n)  OPT  (1/n2 + 1/(n2- 1) +…+ 1)  Õ(log5 n)  OPT

  17. 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 • For every pair si,ti P the path connecting them in T goes through r r

  18. r 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 • Then we could use the O(log n) approximation of [MMP’00]

  19. 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(log2n)  (OPT / h') • Theorem 3: There is an O (log2n) 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(log4n)  (OPT / h') This implies an approximation Õ(log5n) for MC-BB.

  20. 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(log2n)  (OPT / h') • Consider an optimum solution OPT. • Let E* be the edge set of OPT,OPTcbe its costand OPTl be its length.

  21. 5 11 8 12 21 More Details of the Proof of Theorem 2: • Note that OPT may have cycles • By the result of [Elkin, Emek, Spielman, and Tang ’05] on probabilistic distribution on spanning trees and by loosing a factor Õ(log2n) on length, we can assume that E* is a forest T (WLOG we assume T is connected).

  22. More Details of the Proof of Theorem 2: • From T we obtain a collection of rooted subtrees T1,…,Ta 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 Ti; further the root of Ti is their least common ancestor • The total cost of the junction trees is at most Õ(log2n)  OPT (O(log n)  OPTc +Õ(log2n)  OPTl) • Thus at least one of junction trees of T1,…,Ta has the desired density of Õ(log2n)  (OPT / h')

  23. More Details of the Proof of Theorem 2: • Given T, we pick a centeroid r1 (i.e., largest remaining component has at most 2/3 |V(T)| vertices). • Add tree T rooted at r1 to the collection • Remove r1 from Tand apply the procedure recursively to each of the resulting component • Each pair is on exactly one subtree in the collection • Depth of recursion is in O(log n) r

  24. r Some Details of the Proof of Theorem 3: • Theorem 3:There is anO(log2n)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 rootrwith lowest density (= cost of solution / # of pairs in sol). • 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

  25. Some Details of the Proof of Theorem 3: • We solve the LP by setting ys=yt 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 yi 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)

  26. 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(log4n)) [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] • 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(log4n)

  27. 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(log2h) 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(log4h) for general demands (solving an open problem)

  28. 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

  29. Group Cost-Distance A solution connects each group Si using a tree Ti F = union of edges in the trees Ti fe : number of trees Ti using edge e

  30. Group Cost-Distance • By generalizing our current approach, we can obtain an • O(log6n) approximation for this problem • [Gupta, Hajiaghayi, and Kumar ’07].. • It is 2^O(log1-en) hard if each edge has different • grouping [Gupta, Hajiaghayi, and Kumar ’07].

  31. Recent Extensions • The result O(log4n) 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]. • O(log3n) approximation for non-uniform buy at bulk when demands are polynomial [Kortsarz and Nutov’ 07] • O(log4n) approximation when want to have two disjoint paths between each demand pair [Chekuri, Antonakapoulos, Shepherd and Zhang’ 07] • O(n1/2) approximation for generalized directed Steiner tree [Chekuri, Even, Gupta, and Segev’ 08]. • Oblivious network design with ratio O(log3n) for uniformbuy at bulk, i.e., costs of all edges are the same sub-additive function f [Gupta, Hajiaghayi, and Raecke ’07].

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

  33. Thanks for your attention… تشکر Obrigado

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