Boosted Sampling: Approximation Algorithms for Stochastic Problems

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Boosted Sampling: Approximation Algorithms for Stochastic

Problems

Martin Pál

Joint work with Anupam Gupta R. Ravi Amitabh Sinha

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Anctarticast, Inc.

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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) = eSol ce

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Unknown demand?

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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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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 TUsers and all jT

Want compact representation of Sol2 by an algorithm

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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. pj 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 pT. [Our work].

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Example

S1 = { }S2 = { }S3 = { }

…

L

L

σ = 2

OPT = (1+σ) L =3 L

Connected OPT = min(2+σ,2σ) L = 4 L

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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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The Algorithm BS

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 = S1S2 … Sσ.

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

Given Alg, an -approx. for deterministic Steiner Tree

.

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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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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 Si

such 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.

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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’.

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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)

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Cost sharing function

Input: A set of users S’

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

Example: Build a spanning tree on S’ root.

Let ξ(S’,j) = cost of parental edge.

Note:

• jS’ ξ(S’,j) = cost of MST(S’)

• jS’ ξ(S’,j) 2 cost of Steiner(S’)

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Strictness

A csf ξ(,) is β-strict, if

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

for any S,TUsers.

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

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Removing the root

L >> OPT

Problem: Building a Steiner tree on too expensive

Soln: build a Steiner forest on samples {S1, S2,…,Sσ}.

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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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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/σmax accept Si

Let S be the union of accepted Si’s

Output Alg(S) as the first stage solution

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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)

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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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+++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

.

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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&Mirrokni 04]

•Stochastic LP solving&rounding: [Shmoys&Swamy04]

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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

Connectivity problems: satisfied(j) iff j’s terminals connected

Cut problems: satisfied(j) iff j’s terminals disconnected

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Vertex Cover8

3

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10 9

4

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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

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21

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2 3

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4 2

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31

1

2 3

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3 2

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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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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)!

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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.

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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

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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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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.

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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)

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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Special distributions: Rent or BuyStochastic Steiner Network:

client j = pair of terminals sj, tj

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!

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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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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.