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24 Feb 04 Boosted Sampling 1

Boosted Sampling: Approximation Algorithms for Stochastic

Problems

Martin Pál

Joint work withAnupam Gupta R. Ravi

Amitabh Sinha


Infrastructure Design Problems

Build a solution Sol of minimal cost, so that every user is satisfied.

minimize cost(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) = eSol ce


Infrastructure Design Problems

Assumption: Sol is a set of elements

cost(Sol) = elemSol 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


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.


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


The model

Two stage stochastic model with recourse:

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 TUsers and all jT

Want compact representation of Sol2 by an algorithm


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


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]


No restriction on distributions

Previous works often assume special distributions:

•Scenario model: There are k sets of users – scenarios; each scenario Ti has probability pi. [Ravi & Sinha 03].

•Independent decisions model: each client j appears with prob. pj 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).


The Framework

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 solution for clients S = S1S2

… Sσ.3. Actual set T of clients appears. To build second stage solution Sol2, use Alg to augment Sol1 to a feasible solution for T.

Given an approx. algorithm Alg for a deterministic problem:

Example: Steiner Tree


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.


First Stage Cost

Recall: We - sample S1,S2 ,…,Sσ from π.

- use Alg to build solution Sol1 feasible for S=i Si

Lemma: E[cost(Sol1)] α Z*.

Pf: Let Sol = Opt1 [ Opt2(S1) … Opt2(Sσ) ].

E[cost(Sol)] cost(Opt1) + i Eπ[cost(Opt2(Si))]

= Z*.

E[cost(Sol1)] α E[cost(Sol)] (α-approximation).

Opt cost Z* = cost(Opt1) + σ Eπ[cost(Opt2(T))].


Second stage cost

After Stage 2, have a solution for S’ = S1 … Sσ T.

Let Sol’ = Opt1 [ Opt2(S1) … Opt2(Sσ) Opt2(T)].

E[cost(Sol’)] cost(Opt1) + (σ+1) Eπ[cost(Opt2(Si))]

(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 S1 … Sσ.


Idea: cost sharing

Scenario 1:

Pretend to build a solution for S’ = S T.

Charge each jS’ 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: jS’ ξ(S’,j) Opt(S’)

We argued: E[jT ξ(S’,j)] Z*/σ (by symmetry)

Want to prove:

Augmenting cost in Scenario 2 β jT ξ(S’,j)


Cost sharing function

Input: Instance of P and set of users S’

Output: cost share ξ(S,j) of each user jS’

Example: Build a spanning tree on S’ root.

Let ξ(S’,j) = cost of parental edge/2.

Note: - 2 jS’ ξ(S’,j) = cost of MST(S’)

- jS’ ξ(S’,j) cost of Steiner(S’)


What properties of ξ(,) do we need?

(P1) Good approximation: cost(Alg(S)) Opt(S)

(P2) Cost shares do not overpay: jS ξ(S,j) cost(S)

(P3) Strictness: For any S,TUsers:

cost of Augment(Alg(S), T) β jT ξ(S T, j)

Second stage cost = σ cost(Augment(Alg(i Si), T)) σ β jT ξ(j Sj T, j)

E[jT ξ(j Sj T, j)] Z*/σ

Hence, E[second stage cost] σ β Z*/σ = β Z*.


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 jT 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 forStochastic Steiner Tree.


Vertex Cover8

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

11

1

1 1

1

1 1

1

11

1

2 2

1

2 2

2

21

1

2 3

1

4 2

2

31

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


Strictness for Vertex Cover1

1

1

1

1

n+1n+1n

S = blue edges1

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)!


Making Alg strictAlg’: - Run Alg on the same input.

- Buy all vertices that are at least 50% paid for.

1

1

1

1

1

n+1n+1n

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.


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 jT αj’ ≥ ½ cost(v) , then T can pay for ¼ of v in the augmentation step.

•If jS αj ≥ ½ cost(v), then v would be open in Alg(S).

(almost.. need to worry that Alg(S T) and Alg(S) behave differently.)

α1

α2

α3

α1’

α2’Alg(S T) S = blue edges

T = red edgesv


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.


Steiner Network

client j = pair of terminals sj, tj

satisfied(j): sj, tj connected by a path2-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.


The Buy at Bulk problem


Solution: an sj, tj path for j=1,…,n

cost(e) = ce 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


Special distributions: Rent or BuyStochastic Steiner Network:


satisfied(j): sj, tj connected by a path

cost(e) = ce min(1, σ/n #paths using e)# paths using e

costn/σ

Suppose.. π({j}) = 1/n

π(S) = 0 if |S|1

Sol2({j}) is just a path!


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)


What if σ is also stochastic?

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/σmax accept Si

Let S be the union of accepted Si’s

Output Alg(S) as the first stage solution


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)


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?


+++THE++END+++

Note that if π consists of a small number of scenarios, this can be transformed to a deterministic problem.

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 TUsers and all jT

.