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TAM 5.1.'05Boosted Sampling1 Boosted Sampling: Approximation Algorithms for Stochastic Problems Martin Pál Joint work with Anupam Gupta R. RaviAmitabh Sinha

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TAM 5.1.'05Boosted Sampling2 Anctarticast, Inc.

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TAM 5.1.'05Boosted Sampling3 Optimization Problem: Build a solution Sol of minimal cost, so that every user is satisfied. minimizecost(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 c e

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TAM 5.1.'05Boosted Sampling4 Unknown demand? ? ? ? ? ? ? ?

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TAM 5.1.'05Boosted Sampling5 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. drawn from a known distribution π

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TAM 5.1.'05Boosted Sampling6 The model Two stage stochastic model with recourse: Find Sol 1 Edges and Sol 2 : 2 Users 2 Edges to minimizecost(Sol 1 ) + σ E π(T) [cost(Sol 2 (T))] subject to happy(j, Sol 1 Sol 2 (T)) for all sets T Users and all j T Want compact representation of Sol 2 by an algorithm

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TAM 5.1.'05Boosted Sampling7 What distribution? Scenario model: There are k sets of users – scenarios; each scenario T i has probability p i. (e.g.[Ravi & Sinha 03]) Independent decisions model: each client j appears with prob. p j independently of others (e.g.[Immorlica et al 04]) Oracle model: at our request, an oracle gives us a sample T; each set T can have probability p T. [Our work].

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TAM 5.1.'05Boosted Sampling8 Example S 1 = { } S 2 = { } S 3 = { } … L L σ = 2 OPT = (1+σ) L =3 L Connected OPT = min(2+σ,2σ) L = 4 L

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TAM 5.1.'05Boosted Sampling9 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.

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TAM 5.1.'05Boosted Sampling10 The Algorithm BS 1. Boosted Sampling: Draw σ samples of clients S 1,S 2,…,S σ from the distribution π. 2. Build the first stage solution Sol 1 : use Alg to build a tree for clients S = S 1 S 2 … S σ. 3. Actual set T of clients appears. To build second stage solution Sol 2, use Alg to augment Sol 1 to a feasible tree for T. Given Alg, an -approx. for deterministic Steiner Tree.

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TAM 5.1.'05Boosted Sampling11 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

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TAM 5.1.'05Boosted Sampling12 First stage cost Recall: BS builds a tree on samples S 1,S 2,…,S σ from π. Lemma: There is a (random) tree Sol 1 on S= i S i such that E[cost(Sol 1 )] Z *. stochastic optimum cost = Z * = cost(Opt 1 ) + σ E π [cost(Opt 2 (T))]. Lemma: BS pays at most α Z * in the first stage.

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TAM 5.1.'05Boosted Sampling13 Second stage cost After Stage 2, have a tree for S = S 1 … S σ T. There is a cheap tree Sol covering S. Sol = Opt 1 [ Opt 2 (S 1 ) … Opt 2 (S σ ) Opt 2 (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.

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TAM 5.1.'05Boosted Sampling14 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 T ξ(S,j)] Z * /σ (by symmetry) Want to prove: Augmenting cost in Scenario 2 β j T ξ(S,j)

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TAM 5.1.'05Boosted Sampling15 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)

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TAM 5.1.'05Boosted Sampling16 Strictness A csf ξ(, ) is β-strict, if cost of Augment(Alg(S), T) β j T ξ(S T, j) for any S,T Users. Second stage cost = σ cost(Augment(Alg( i S i ), T)) σ β j T ξ( j S j T, j) Fact: E[shares of T] Z * /σ Hence: E[second stage cost] σ β Z * /σ = β Z *. S T

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TAM 5.1.'05Boosted Sampling17 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 ξ is a 2-strict cost sharing function for Alg. Theorem: We have a 4-approx for Stochastic Steiner Tree.

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TAM 5.1.'05Boosted Sampling18 Removing the root L >> OPT Problem: Building a Steiner tree on too expensive Soln: build a Steiner forest on samples {S 1, S 2,…, S σ }.

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TAM 5.1.'05Boosted Sampling19 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)

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TAM 5.1.'05Boosted Sampling20 What if σ is random? Suppose σ is also a random variable. π(S, σ) – joint distribution For i=1, 2, …, σ max do sample (S i, σ i ) from π with prob. σ i /σ max accept S i Let S be the union of accepted S i s Output Alg(S) as the first stage solution

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TAM 5.1.'05Boosted Sampling21 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 σ 1. On Wednesday, clients finally show up. Edges now cost σ 2 (σ 2 >σ 1 ), 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)

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TAM 5.1.'05Boosted Sampling22 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?

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TAM 5.1.'05Boosted Sampling23 +++THE++END+++ Note that if π consists of a small number of scenarios, this can be transformed to a deterministic problem. Find Sol 1 Elems and Sol 2 : 2 Users 2 Elems to minimizecost(Sol 1 ) + σ E π(T) [cost(Sol 2 (T))] subject to satisfied(j, Sol 1 Sol 2 (T)) for all sets T Users and all j T.

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TAM 5.1.'05Boosted Sampling24 Related work Stochastic linear programming dates back to works of Dantzig, Beale in the mid-50s 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&Mirrokni 04] Stochastic LP solving&rounding: [Shmoys&Swamy04]

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TAM 5.1.'05Boosted Sampling25 Infrastructure Design Problems Assumption: Sol is a set of elements cost(Sol) = elem Sol cost(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 js terminals connected Cut problems: satisfied(j) iff js terminals disconnected

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TAM 5.1.'05Boosted Sampling26 Vertex Cover 8 3 3 10 9 4 5 Users: edges Solution: Set of vertices that covers all edges Edge {uv} covered if at least one of u,v picked. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 2 2 2 2 1 1 2 3 1 4 2 2 3 1 1 2 3 1 3 2 2 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

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TAM 5.1.'05Boosted Sampling27 Strictness for Vertex Cover 1 1 1 1 1 n+1 n S = blue edges 1 1 1 1 1 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)!

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TAM 5.1.'05Boosted Sampling28 Making Alg strict Alg: - Run Alg on the same input. - Buy all vertices that are at least 50% paid for. 1 1 1 1 1 n+1 n 1 1 1 1 1 ½ of each vertex paid for, each edge paying for two vertices still a 4-approximation. Augmentation (at least in our example) is free.

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TAM 5.1.'05Boosted Sampling29 Why should strictness hold? Alg: - Run Alg on the same input. - Buy all vertices that are at least 50% paid for. Suppose vertex v fully paid for in Alg(S T). If j T α j ½ cost(v), then T can pay for ¼ of v in the augmentation step. If j S α j ½ cost(v), then v would be open in Alg(S). (almost.. need to worry that Alg(S T) and Alg(S) behave differently.) α1α1 α2α2 α3α3 α 1 α 2 Alg(S T) S = blue edges T = red edges v

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TAM 5.1.'05Boosted Sampling30 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.

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TAM 5.1.'05Boosted Sampling31 Steiner Network client j = pair of terminals s j, t j satisfied(j): s j, t j 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.

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TAM 5.1.'05Boosted Sampling32 The Buy at Bulk problem client j = pair of terminals s j, t j Solution: an s j, t j path for j=1,…,n cost(e) = c e f(# paths using e) cost # paths using e f(e): # paths using e cost Rent or Buy: two pipes Rent: $1 per path Buy: $M, unlimited # of paths

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TAM 5.1.'05Boosted Sampling33 Special distributions: Rent or Buy Stochastic Steiner Network: client j = pair of terminals s j, t j satisfied(j): s j, t j connected by a path cost(e) = c e min(1, σ/n #paths using e) # paths using e cost n/σn/σ Suppose.. π({j}) = 1/n π(S) = 0 if |S| 1 Sol 2 ({j}) is just a path!

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TAM 5.1.'05Boosted Sampling34 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)

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TAM 5.1.'05Boosted Sampling35 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.

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