24 Feb 04Boosted Sampling1 Boosted Sampling: Approximation Algorithms for Stochastic Problems Martin Pál Joint work with Anupam Gupta R. Ravi Amitabh Sinha

24 Feb 04Boosted Sampling2 Infrastructure Design Problems Build a solution Sol of minimal cost, so that every user is satisfied. minimizecost(Sol) subject to satisfied(j,Sol)for j=1, 2, …, n For example, Steiner tree: Sol: set of edges to build satisfied(j,Sol) iff there is a path from terminal j to root cost(Sol) =  e  Sol c e

24 Feb 04Boosted Sampling3 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 Steiner network: satisfied(j) iff j’s terminals connected by a path Cut problems: satisfied(j) iff j’s terminals disconnected

24 Feb 04Boosted Sampling4 Dealing with uncertainity Often, we do not know the exact requirements of users. Building in advance reduces cost – but we do not have enough information. As time progresses, we gain more information about the demands – but building under time pressure is costly. Tradeoff between information and cost.

24 Feb 04Boosted Sampling5 The model Two stage stochastic model with recourse: On Monday, elements are cheap, but we do not know how many/which clients will show up. We can buy some elements. On Tuesday, clients show up. Elements are now more expensive (by an inflation factor σ). We have to buy more elements to satisfy all clients. drawn from a known distribution π

24 Feb 04Boosted Sampling6 The model Two stage stochastic model with recourse: 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 Want compact representation of Sol 2 by an algorithm

24 Feb 04Boosted Sampling7 Related work Stochastic linear programming dates back to works of Dantzig, Beale in the mid-50’s Only moderate progress on stochastic IP/MIP Scheduling literature, various distributions of job lengths Single stage stochastic: maybecast [Karger&Minkoff00], bursty connections [Kleinberg,Rabani&Tardos00] … Stochastic versions of NP-hard problems (restricted π) [Ravi & Sinha 03], [Immorlica, Karger, Minkoff & Mirrokni 04] Extensive literature on each deterministic problem

24 Feb 04Boosted Sampling8 Our work We propose a simple but powerful framework to find approximate solutions to two stage stochastic problems using approximation algorithms for their deterministic counterparts. For a number of problems, including Steiner Tree, Facility Location, Single Sink Rent or Buy and Steiner Forest (weaker model) our framework gives constant approximation. Analysis is based on strict cost sharing, developed by [Gupta,Kumar,P.&Roughgarden03]

24 Feb 04Boosted Sampling9 No restriction on distributions Previous works often assume special distributions: Scenario model: There are k sets of users – scenarios; each scenario T i has probability p i. [Ravi & Sinha 03]. Independent decisions model: each client j appears with prob. p j independently of others [Immorlica et al 04]. In contrast, our scheme works for arbitrary distributions (although the independent coinflips model sometimes allows us to prove improved guarantees).

24 Feb 04Boosted Sampling10 The Framework 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 solution 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 solution for T. Given an approx. algorithm Alg for a deterministic problem: Example: Steiner Tree

24 Feb 04Boosted Sampling11 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.

24 Feb 04Boosted Sampling12 First Stage Cost Recall: We - sample S 1,S 2,…,S σ from π. - use Alg to build solution Sol 1 feasible for S=  i S i Lemma: E[cost(Sol 1 )]  α  Z *. Pf: Let Sol = Opt 1  [ Opt 2 (S 1 )  …  Opt 2 (S σ ) ]. E[cost(Sol)]  cost(Opt 1 ) +  i E π [cost(Opt 2 (S i ))] = Z *. E[cost(Sol 1 )]  α  E[cost(Sol)] (α-approximation). Opt cost Z * = cost(Opt 1 ) + σ E π [cost(Opt 2 (T))].

24 Feb 04Boosted Sampling13 Second stage cost After Stage 2, have a solution for S’ = S 1  …  S σ  T. Let Sol’ = Opt 1  [ Opt 2 (S 1 )  …  Opt 2 (S σ )  Opt 2 (T)]. E[cost(Sol’)]  cost(Opt 1 ) + (σ+1) E π [cost(Opt 2 (S i ))]  (σ+1)/σ  Z *. T is “responsible” for 1/(σ+1) part of Sol’. If built in Stage 1, it would cost  Z * /σ. Need to build it in Stage 2  pay Z *. Problem: do not T know when building a solution for S 1  …  S σ.

24 Feb 04Boosted 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)

24 Feb 04Boosted Sampling15 Cost sharing function Input: Instance of P and set of users S’ Output: cost share ξ(S,j) of each user j  S’ Example: Build a spanning tree on S’  root. Let ξ(S’,j) = cost of parental edge/2. Note: - 2   j  S’ ξ(S’,j) = cost of MST(S’) -  j  S’ ξ(S’,j)  cost of Steiner(S’)

24 Feb 04Boosted Sampling16 What properties of ξ( ,  ) do we need? (P1) Good approximation: cost(Alg(S))   Opt(S) (P2) Cost shares do not overpay:  j  S ξ(S,j)  cost(S) (P3) Strictness: For any S,T  Users: cost of Augment(Alg(S), T)  β   j  T ξ(S  T, j) Second stage cost = σ cost(Augment(Alg(  i S i ), T))   σ β   j  T ξ(  j S j  T, j) E[  j  T ξ(  j S j  T, j)]  Z * /σ Hence, E[second stage cost]  σ β  Z * /σ = β  Z *.

24 Feb 04Boosted Sampling17 Strictness for Steiner Tree Alg(S) = Min-cost spanning tree MST(S) ξ(S,j) = cost of parental edge/2 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.

24 Feb 04Boosted Sampling18 Vertex Cover Users: edges Solution: Set of vertices that covers all edges Edge {uv} covered if at least one of u,v picked 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

24 Feb 04Boosted Sampling19 Strictness for Vertex Cover n+1 n 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)!

24 Feb 04Boosted Sampling20 Making Alg strict Alg’: - Run Alg on the same input. - Buy all vertices that are at least 50% paid for n+1 n ½ of each vertex paid for, each edge paying for two vertices  still a 4-approximation. Augmentation (at least in our example) is free.

24 Feb 04Boosted Sampling21 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 α1’α1’ α2’α2’ Alg(S  T) S = blue edges T = red edges v

24 Feb 04Boosted Sampling22 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.

24 Feb 04Boosted Sampling23 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.

24 Feb 04Boosted Sampling24 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

24 Feb 04Boosted Sampling25 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!

24 Feb 04Boosted Sampling26 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)

24 Feb 04Boosted Sampling27 What if σ is also stochastic? 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

24 Feb 04Boosted Sampling28 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)

24 Feb 04Boosted Sampling29 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 strictness for Steiner Forest – so far we have only uni-strictness. Cut problems: Can we say anything about Multicut? Single- source multicut?

24 Feb 04Boosted Sampling30 +++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.