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Boosted Sampling: Approximation Algorithms for Stochastic Problems

Boosted Sampling: Approximation Algorithms for Stochastic Problems. Martin P ál Joint work with Anupam Gupta R. Ravi Amitabh Sinha. Anctarticast, Inc. Optimization Problem:. Build a solution Sol of minimal cost, so that every user is satisfied. minimize cost( Sol )

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Boosted Sampling: Approximation Algorithms for Stochastic Problems

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  1. Boosted Sampling: Approximation Algorithms for Stochastic Problems Martin Pál Joint work with Anupam Gupta R. Ravi Amitabh Sinha Boosted Sampling

  2. Anctarticast, Inc. Boosted Sampling

  3. Optimization Problem: Build a solution Sol of minimal cost, so that every user is satisfied. minimize cost(Sol) subject to happy(j,Sol) for j=1, 2, …, n For example, Steiner tree: Sol: set of links to build happy(j,Sol) iff there is a path from terminal j to root cost(Sol) = eSol ce Boosted Sampling

  4. ? ? ? ? ? ? ? Unknown demand? Boosted Sampling

  5. drawn from a known distribution π The model Two stage stochastic model with recourse: On Monday, links are cheap, but we do not know how many/which clients will show up. We can buy some links. On Tuesday, clients appear. Links are now σ times more expensive. We have to buy enough links to satisfy all clients. Boosted Sampling

  6. Want compact representation of Sol2 by an algorithm The model Two stage stochastic model with recourse: Find Sol1 Edges and Sol2 : 2Users 2Edges to minimize cost(Sol1) + σ Eπ(T)[cost(Sol2(T))] subject to happy(j, Sol1  Sol2(T)) for all sets TUsers and all jT Boosted Sampling

  7. What distribution? • Scenario model: There are k sets of users – scenarios; each scenario Ti has probability pi. (e.g.[Ravi & Sinha 03]) • Independent decisions model: each client j appears with prob. pjindependently of others(e.g.[Immorlica et al04]) • Oracle model: at our request, an oracle gives us a sample T; each set T can have probability pT. [Our work]. Boosted Sampling

  8. Example L OPT = (1+σ) L =3 L L Connected OPT = min(2+σ,2σ) L = 4 L S1= { } S2= { } S3= { } … σ = 2 Boosted Sampling

  9. Deterministc Steiner tree cost(Steiner)  cost(MST) cost(MST)  2 cost(Steiner) Theorem: Finding a Minimum Spanning Tree is a 2-approximation algorithm for Minimum Steiner Tree. Boosted Sampling

  10. . The Algorithm BS Given Alg, an-approx. for deterministic Steiner Tree 1. Boosted Sampling: Draw σ samples of clients S1,S2 ,…,Sσ from the distribution π. 2. Build the first stage solution Sol1: use Alg to build a tree for clients S = S1S2  … Sσ. 3. Actual set T of clients appears. To build second stage solution Sol2, use Alg to augmentSol1 to a feasible tree for T. Boosted Sampling

  11. Is BS any good? • Theorem: Boosted sampling algorithm is a 4-approximation for Stochastic Steiner Tree (assuming Alg is a 2-approx..). • Nothing special about Steiner Tree; BS works for other problems: Facility Location, Steiner Network, Vertex Cover.. • Idea: Bound stage costs separately • First stage cheap, because not too many samples, and Alg is good • Second stage is cheap, because samples “dense enough” Boosted Sampling

  12. First stage cost Recall: BS builds a tree on samples S1,S2 ,…,Sσ from π. Lemma: There is a (random) tree Sol1 on S=i Sisuch that E[cost(Sol1)]  Z*. stochastic optimum cost = Z* = cost(Opt1) + σ Eπ[cost(Opt2(T))]. Lemma: BS pays at most αZ*in the first stage. Boosted Sampling

  13. Second stage cost After Stage 2, have a tree for S’ = S1 …Sσ T. There is a cheap tree Sol’ covering S’. Sol’ = Opt1 [Opt2(S1)…Opt2(Sσ)Opt2(T)]. Fact: E[cost(Sol’)]  (σ+1)/σ  Z*. T is “responsible” for 1/(σ+1) part of Sol’. At Stage 1 costs, it would pay Z*/σ. Need to pay Stage 2 premium  pay Z*. Problem: do not know T when building Sol’. Boosted Sampling

  14. Idea: cost sharing Scenario 1: Pretend to build a solution for S’ = S T. Charge each jS’ some amount ξ(S’,j). Scenario 2: Build a solution Alg(S) for S. Augment Alg(S) to a valid solution for S’ = S T. Assume: jS’ξ(S’,j) Opt(S’) We argued: E[jTξ(S’,j)] Z*/σ (by symmetry) Want to prove: Augmenting cost in Scenario 2  β  jTξ(S’,j) Boosted Sampling

  15. Cost sharing function • Input: A set of users S’ • Output: cost share ξ(S,j) for each user jS’ • Example: Build a spanning tree on S’ root. • Let ξ(S’,j) = cost of parental edge. • Note: • jS’ξ(S’,j) = cost of MST(S’) • jS’ξ(S’,j) 2 cost of Steiner(S’) Boosted Sampling

  16. Strictness A csf ξ(,) is β-strict, if cost of Augment(Alg(S), T)  β  jTξ(S T, j) for any S,TUsers. S Second stage cost = σ cost(Augment(Alg(i Si), T))   σβ  jTξ(j SjT, j) Fact: E[shares of T]  Z*/σ Hence:E[second stage cost]  σβ  Z*/σ = β  Z*. T Boosted Sampling

  17. Strictness for Steiner Tree Alg(S) = Min-cost spanning tree MST(S) ξ(S,j) =cost of parental edge in MST(S) Augment(Alg(S), T): for all jT build its parental edge in MST(S T) Alg is a 2-approx for Steiner Tree ξis a 2-strict cost sharing function for Alg.Theorem: We have a 4-approx for Stochastic Steiner Tree. Boosted Sampling

  18. Problem: Building a Steiner tree on too expensive Removing the root L >> OPT Soln: build a Steiner forest on samples {S1, S2,…,Sσ}. Boosted Sampling

  19. Works for other problems, too BS works for any problem that is - subadditive (union of solns for S,T is a soln for S T) - has α-approx algo that admits β-strict cost sharing Constant approximation algorithms for stochastic Facility Location, Vertex Cover, Steiner Network.. (hard part: prove strictness) Boosted Sampling

  20. What if σ is random? Suppose σ is also a random variable. π(S, σ) – joint distribution For i=1, 2, …, σmax do sample (Si, σi) from π with prob. σi/σmaxacceptSi Let S be the union of accepted Si’s Output Alg(S) as the first stage solution Boosted Sampling

  21. Multistage problems • Three stage stochastic Steiner Tree: • On Monday, edges cost 1. We only know the probability distribution π. • On Tuesday, results of a market survey come in. We gain some information I, and update π to the conditional distribution π|I. Edges cost σ1. • On Wednesday, clients finally show up. Edges now cost σ2 (σ2>σ1), and we must buy enough to connect all clients. • Theorem: There is a 6-approximation for three stage stochastic Steiner Tree (in general, 2k approximation for k stage problem) Boosted Sampling

  22. Conclusions • We have seen a randomized algorithm for a stochastic problem: using sampling to solve problems involving randomness. • Do we need strict cost sharing? Our proof requires strictness – maybe there is a weaker property? Maybe we can prove guarantees for arbitrary subadditive problems? • Prove full strictness for Steiner Forest – so far we have only uni-strictness. • Cut problems: Can we say anything about Multicut? Single-source multicut? Boosted Sampling

  23. Find Sol1 Elems and Sol2 : 2Users 2Elems to minimize cost(Sol1) + σ Eπ(T)[cost(Sol2(T))] subject to satisfied(j, Sol1  Sol2(T)) for all sets TUsers and all jT +++THE++END+++ Note that if π consists of a small number of scenarios, this can be transformed to a deterministic problem. . Boosted Sampling

  24. Related work • Stochastic linear programming dates back to works of Dantzig, Beale in the mid-50’s • Scheduling literature, various distributions of job lengths • Resource installation[Dye,Stougie&Tomasgard03] • Single stage stochastic: maybecast [Karger&Minkoff00], bursty connections [Kleinberg,Rabani&Tardos00]… • Stochastic versions of NP-hard problems (restricted π)[Ravi&Sinha03], [Immorlica,Karger,Minkoff&Mirrokni04] • Stochastic LP solving&rounding: [Shmoys&Swamy04] Boosted Sampling

  25. Infrastructure Design Problems Assumption: Sol is a set of elements cost(Sol) = elemSolcost(elem) Facility location: satisfied(j) iff j connected to an open facility Vertex Cover: satisfied(e={uv}) iff u or v in the cover Connectivity problems: satisfied(j) iff j’s terminals connected Cut problems: satisfied(j) iff j’s terminals disconnected Boosted Sampling

  26. 2 2 2 1 2 2 2 1 2 3 3 1 2 3 4 1 2 2 2 1 1 1 1 1 3 2 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Vertex Cover 8 Users: edges Solution: Set of vertices that covers all edges Edge {uv} covered if at least one of u,v picked. 4 3 10 9 5 3 Alg: Edges uniformly raise contributions Vertex can be paid for by neighboring edges  freeze all edges adjacent to it. Buy the vertex. Edges may be paying for both endpoints  2-approximation Natural cost shares: ξ(S, e) = contribution of e Boosted Sampling

  27. S = blue edges T = red edge • Alg(S) = blue vertices: • Augment(Alg(S), T) costs (n+1) • ξ(S T, T) =1 • Find a better ξ? Do not know how. Instead, make Alg(S) buy a center vertex.  gap Ω(n)! Strictness for Vertex Cover 1 1 1 1 n+1 n+1 1 1 n 1 1 1 1 Boosted Sampling

  28. Augmentation (at least in our example) is free. Making Alg strict Alg’: - Run Alg on the same input. - Buy all vertices that are at least 50% paid for. 1 1 1 1 n+1 n+1 1 1 n 1 1 1 1 ½ of each vertex paid for, each edge paying for two vertices  still a 4-approximation. Boosted Sampling

  29. Why should strictness hold? Alg’: - Run Alg on the same input. - Buy all vertices that are at least 50% paid for. α1 α1’ S = blue edges T = red edges v Alg(S T) α2 α2’ α3 • Suppose vertex v fully paid for in Alg(S T). • If jTαj’ ≥ ½ cost(v), then T can pay for ¼ of v in the augmentation step. • If jSαj≥ ½ cost(v), then v would be open in Alg(S). • (almost.. need to worry that Alg(S T) and Alg(S) behave differently.) Boosted Sampling

  30. Metric facility location Input: a set of facilities and a set of cities living in a metric space. Solution: Set of open facilities, a path from each city to an open facility. “Off the shelf” components: 3-approx. algorithm [Mettu&Plaxton00]. Turns out that cost sharing fn [P.&Tardos03] is 5.45 strict. Theorem: There is a 8.45-approx for stochastic FL. Boosted Sampling

  31. Steiner Network client j = pair of terminals sj, tj satisfied(j): sj, tj connected by a path 2-approximation algorithms known ([Agarwal,Klein&Ravi91], [Goemans&Williamson95]), but do not admit strict cost sharing. [Gupta,Kumar,P.,Roughgarden03]: 4-approx algorithm that admits 4-uni-strict cost sharing Theorem: 8-approx for Stochastic Steiner Network in the “independent coinflips” model. Boosted Sampling

  32. cost f(e): # paths using e cost # paths using e The Buy at Bulk problem client j = pair of terminals sj, tj Solution: an sj, tj path for j=1,…,n cost(e) = ce f(# paths using e) Rent or Buy: two pipes Rent: $1 per path Buy: $M, unlimited # of paths Boosted Sampling

  33. cost n/σ cost(e) = ce min(1, σ/n #paths using e) # paths using e Special distributions: Rent or Buy Stochastic Steiner Network: client j = pair of terminals sj, tj satisfied(j): sj, tj connected by a path Suppose.. π({j}) = 1/n π(S) = 0 if |S|1 Sol2({j}) is just a path! Boosted Sampling

  34. Rent or Buy The trick works for any problem P. (can solve Rent-or-Buy Vertex Cover,..) These techniques give the best approximation for Single-Sink Rent-or-Buy (3.55 approx [Gupta,Kumar,Roughgarden03]), and Multicommodity Rent or Buy (8-approx [Gupta,Kumar,P.,Roughgarden03], 6.83-approx [Becchetti, Konemann, Leonardi,P.04]). “Bootstrap” to stochastic Rent-or-Buy: - 6 approximation for Stochastic Single-Sink RoB - 12 approx for Stochastic Multicommodity RoB (indep. coinflips) Boosted Sampling

  35. Performance Guarantee Theorem:Let P be a sub-additive problem, with α-approximation algorithm, that admits β-strict cost sharing. Stochastic(P) has (α+β) approx.  Corollary: Stochastic Steiner Tree, Facility Location, Vertex Cover, Steiner Network (restricted model)… have constant factor approximation algorithms. Corollary: Deterministic and stochastic Rent-or-Buy versions of these problems have constant approximations. Boosted Sampling

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