approximation algorithms for stochastic combinatorial optimization r. ravi carnegie mellon...

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Approximation Algorithms Approximation Algorithms for Stochastic for Stochastic Combinatorial Combinatorial Optimization Optimization R. Ravi R. Ravi Carnegie Mellon University Carnegie Mellon University Joint work with: Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere, CMU Anupam Gupta, CMU Anupam Gupta, CMU Martin Pal, DIMACS Martin Pal, DIMACS Mohit Singh, CMU Mohit Singh, CMU Amitabh Sinha, U. Michigan Amitabh Sinha, U. Michigan Sources: [RS IPCO 04, GPRS STOC 04, GRS Sources: [RS IPCO 04, GPRS STOC 04, GRS FOCS 04, DRS IPCO 05] FOCS 04, DRS IPCO 05]

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Page 1: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

Approximation Algorithms for Approximation Algorithms for Stochastic Combinatorial Stochastic Combinatorial

OptimizationOptimizationR. RaviR. Ravi

Carnegie Mellon UniversityCarnegie Mellon University

Joint work with:Joint work with: Kedar Dhamdhere, CMUKedar Dhamdhere, CMU Anupam Gupta, CMUAnupam Gupta, CMU Martin Pal, DIMACSMartin Pal, DIMACS Mohit Singh, CMUMohit Singh, CMU Amitabh Sinha, U. MichiganAmitabh Sinha, U. Michigan

Sources: [RS IPCO 04, GPRS STOC 04, GRS FOCS 04, Sources: [RS IPCO 04, GPRS STOC 04, GRS FOCS 04, DRS IPCO 05]DRS IPCO 05]

Page 2: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

22

OutlineOutline

Motivation: The cable company Motivation: The cable company problemproblem

Model and literature reviewModel and literature review Solution to the cable company Solution to the cable company

problemproblem General covering problemGeneral covering problem Scenario dependent cost modelScenario dependent cost model

Page 3: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

33

The cable company problemThe cable company problem

Cable company Cable company plans to enter a plans to enter a new areanew area

Currently, low Currently, low populationpopulation

Wants to install Wants to install cable infrastructure cable infrastructure in anticipation of in anticipation of future demandfuture demand

Page 4: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

44

The cable company problemThe cable company problem

Future demand Future demand unknown, yet cable unknown, yet cable company needs to company needs to build nowbuild now

Where should Where should cable company cable company install cables?install cables?

Page 5: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

55

The cable company problemThe cable company problem

Future demand Future demand unknown, yet cable unknown, yet cable company needs to company needs to build nowbuild now

Where should Where should cable company cable company install cables?install cables?

Page 6: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

66

The cable company problemThe cable company problem

Future demand Future demand unknown, yet cable unknown, yet cable company needs to company needs to build nowbuild now

Where should Where should cable company cable company install cables?install cables?

Page 7: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

77

The cable company problemThe cable company problem

Future demand Future demand unknown, yet cable unknown, yet cable company needs to company needs to build nowbuild now

Where should Where should cable company cable company install cables?install cables?

Page 8: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

88

The cable company problemThe cable company problem

Future demand Future demand unknown, yet cable unknown, yet cable company needs to company needs to build nowbuild now

ForecastsForecasts of possible of possible future demands future demands existexist

Where should cable Where should cable company install company install cables?cables?

Page 9: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

99

The cable company problemThe cable company problem

Future demand Future demand unknown, yet cable unknown, yet cable company needs to company needs to build nowbuild now

ForecastsForecasts of possible of possible future demands future demands existexist

Where should cable Where should cable company install company install cables?cables?

Page 10: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

1010

The cable company problemThe cable company problem

Future demand Future demand unknown, yet cable unknown, yet cable company needs to company needs to build nowbuild now

ForecastsForecasts of possible of possible future demands future demands existexist

Where should cable Where should cable company install company install cables?cables?

Page 11: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

1111

The cable company problemThe cable company problem

Future demand Future demand unknown, yet cable unknown, yet cable company needs to company needs to build nowbuild now

Forecasts Forecasts of possible of possible future demands future demands existexist

Where should cable Where should cable company install company install cables?cables?

Page 12: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

1212

The cable company problemThe cable company problem

cable company cable company wants to use wants to use demand forecasts, todemand forecasts, to

MinimizeMinimize

Today’s install. Today’s install. costscosts

+ Expected future + Expected future costscosts

Page 13: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

1313

OutlineOutline

Motivation: The cable company Motivation: The cable company problemproblem

Model and literature reviewModel and literature review Solution to the cable company Solution to the cable company

problemproblem General covering problemGeneral covering problem Scenario dependent cost model Scenario dependent cost model

Page 14: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

1414

Stochastic optimizationStochastic optimization

Classical optimization assumed Classical optimization assumed deterministic inputsdeterministic inputs

Page 15: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

1515

Stochastic optimizationStochastic optimization

Classical optimization assumed Classical optimization assumed deterministic inputsdeterministic inputs

Need for modeling Need for modeling data uncertaintydata uncertainty quickly realized [Dantzig ‘55, Beale quickly realized [Dantzig ‘55, Beale ‘61]‘61]

Page 16: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

1616

Stochastic optimizationStochastic optimization

Classical optimization assumes Classical optimization assumes deterministic inputsdeterministic inputs

Need for modeling data uncertainty Need for modeling data uncertainty quickly realized [Dantzig ‘55, Beale quickly realized [Dantzig ‘55, Beale ‘61]‘61]

[Birge, Louveaux ’97, Klein Haneveld, [Birge, Louveaux ’97, Klein Haneveld, van der Vlerk ’99]van der Vlerk ’99]

Page 17: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

1717

ModelModel

Two-stage stochastic opt. with Two-stage stochastic opt. with recourserecourse

Page 18: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

1818

ModelModel

Two-stageTwo-stage stochastic opt. with stochastic opt. with recourserecourse

Two stages of decision making, with Two stages of decision making, with limited information in first stagelimited information in first stage

Page 19: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

1919

ModelModel

Two-stage Two-stage stochastic opt.stochastic opt. with with recourserecourse

Two stages of decision makingTwo stages of decision making Probability distribution governing Probability distribution governing

second-stage data and costs given in second-stage data and costs given in 1st stage1st stage

Page 20: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

2020

ModelModel

Two-stage stochastic opt. with Two-stage stochastic opt. with recourserecourse

Two stages of decision makingTwo stages of decision making Probability dist. governing data and Probability dist. governing data and

costscosts Solution can always be made feasible Solution can always be made feasible

in second stagein second stage

Page 21: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

2121

Mathematical modelMathematical model

ΩΩ: probability space of 2: probability space of 2ndnd stage data stage data

)()(,

)()().().(

.

)]()([.min

PyPx

hyWxT

bxA

ycExc

Page 22: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

2222

Mathematical modelMathematical model

ΩΩ: probability space of 2: probability space of 2ndnd stage data stage data

Extensive form: Enumerate over all Extensive form: Enumerate over all ωω єє ΩΩ

)()(,

)()().().(

.

)]()([.min

PyPx

hyWxT

bxA

ycExc

Page 23: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

2323

Scenario modelsScenario models

Enumerating over all Enumerating over all ωω єє ΩΩ may lead may lead to to very large problem sizevery large problem size

Enumeration (or even approximation) Enumeration (or even approximation) may not be possiblemay not be possible for continuous for continuous domainsdomains

Page 24: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

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New model: Sampling New model: Sampling AccessAccess

““Black boxBlack box” available which ” available which generates a generates a sample sample of 2of 2ndnd stage data stage data with with samesame distributiondistribution as actual 2 as actual 2ndnd stagestage

Bare minimum requirement on model Bare minimum requirement on model of stochastic processof stochastic process

Page 25: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

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Computational complexityComputational complexity

Stochastic optimization problems Stochastic optimization problems solved using Mixed Integer Program solved using Mixed Integer Program formulationsformulations

Solution times prohibitiveSolution times prohibitive NP-hardnessNP-hardness inherentinherent toto problemproblem, not , not

formulation: E.g., 2-stage stochastic formulation: E.g., 2-stage stochastic versions of MST, Shortest paths are versions of MST, Shortest paths are NP-hard.NP-hard.

Page 26: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

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Our goalOur goal

Approximation algorithm using Approximation algorithm using sampling accesssampling access cable company problem cable company problem General model – extensions to other General model – extensions to other

problemsproblems

Page 27: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

2727

Our goalOur goal

Approximation algorithm using Approximation algorithm using sampling accesssampling access cable company problem cable company problem (General model – extensions to other (General model – extensions to other

problems)problems) ConsequencesConsequences

Provable guarantees on solution qualityProvable guarantees on solution quality Minimal requirements of stochastic Minimal requirements of stochastic

processprocess

Page 28: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

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Previous workPrevious work Scheduling with stochastic dataScheduling with stochastic data

Substantial work on exact algorithms [Pinedo ’95]Substantial work on exact algorithms [Pinedo ’95] Some recent approximation algorithms [Goel, Indyk Some recent approximation algorithms [Goel, Indyk

’99; Möhring, Schulz, Uetz ’99]’99; Möhring, Schulz, Uetz ’99] Approximation algorithms for stochastic modelsApproximation algorithms for stochastic models

Resource provisioning with polynomial scenarios Resource provisioning with polynomial scenarios [Dye, Stougie, Tomasgard Nav. Res. Qtrly ’03][Dye, Stougie, Tomasgard Nav. Res. Qtrly ’03]

””Maybecast” Steiner tree: Maybecast” Steiner tree: OO(log (log nn) approximation ) approximation when terminals activate independently [Immorlica, when terminals activate independently [Immorlica, Karger, Minkoff, Mirrokni ’04] Karger, Minkoff, Mirrokni ’04]

Page 29: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

2929

Our workOur work

Approximation algorithms for two-stage Approximation algorithms for two-stage stochastic combinatorial optimizationstochastic combinatorial optimization Polynomial Scenarios model, several problems using LP Polynomial Scenarios model, several problems using LP

rounding, incl. Vertex Cover, Facility Location, Shortest rounding, incl. Vertex Cover, Facility Location, Shortest paths [R., Sinha, July ’03, appeared IPCO ’04]paths [R., Sinha, July ’03, appeared IPCO ’04]

Black-box model: Boosted sampling algorithm for Black-box model: Boosted sampling algorithm for covering problems with subadditivity – general covering problems with subadditivity – general approximation algorithm [Gupta, Pal, R., Sinha STOC ’04]approximation algorithm [Gupta, Pal, R., Sinha STOC ’04]

Steiner trees and network design problems: Polynomial Steiner trees and network design problems: Polynomial scenarios model, Combination of LP rounding and scenarios model, Combination of LP rounding and Primal-Dual [Gupta, R., Sinha FOCS ’04]Primal-Dual [Gupta, R., Sinha FOCS ’04]

Stochastic MSTs under scenario model and Black-box Stochastic MSTs under scenario model and Black-box model with polynomially bounded cost inflations model with polynomially bounded cost inflations [Dhamdhere, R., Singh, To appear, IPCO ’05][Dhamdhere, R., Singh, To appear, IPCO ’05]

Page 30: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

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Related workRelated work

Approximation algorithms for Stochastic Approximation algorithms for Stochastic Combinatorial ProblemsCombinatorial Problems Vertex cover and Steiner trees in restricted models Vertex cover and Steiner trees in restricted models

studied by [Immorlica, Karger, Minkoff, Mirrokni SODA ’04]studied by [Immorlica, Karger, Minkoff, Mirrokni SODA ’04] Rounding for stochastic Set Cover, FPRAS for #P hard Rounding for stochastic Set Cover, FPRAS for #P hard

Stochastic Set Cover LPs [Shmoys, Swamy FOCS ’04]Stochastic Set Cover LPs [Shmoys, Swamy FOCS ’04] Multi-stage stochastic Steiner trees [Hayrapetyan, Swamy, Multi-stage stochastic Steiner trees [Hayrapetyan, Swamy,

Tardos SODA ‘05]Tardos SODA ‘05] Multi-stage Stochastic Set Cover [Shmoys, Swamy, Multi-stage Stochastic Set Cover [Shmoys, Swamy,

manuscript ’04]manuscript ’04] Multi-stage black box model – Extension of Boosted Multi-stage black box model – Extension of Boosted

sampling with rejection [Gupta, Pal, R., Sinha manuscript sampling with rejection [Gupta, Pal, R., Sinha manuscript ’05]’05]

Page 31: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

3131

OutlineOutline

Motivation: The cable company Motivation: The cable company problemproblem

Model and literature reviewModel and literature review Solution to the cable company Solution to the cable company

problemproblem General covering problemGeneral covering problem Scenario dependent cost model Scenario dependent cost model

Page 32: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

3232

The cable company problemThe cable company problem

Cable company Cable company wants to install wants to install cables to serve cables to serve future demandfuture demand

Page 33: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

3333

The cable company problemThe cable company problem

Cable company Cable company wants to install wants to install cables to serve cables to serve future demandfuture demand

Future demand Future demand stochastic, cables stochastic, cables get expensive next get expensive next yearyear

What cables to What cables to install this year?install this year?

Page 34: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

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Steiner Tree - BackgroundSteiner Tree - Background

Graph Graph G=(V,E,c)G=(V,E,c) Terminals Terminals SS, root , root rrSS Steiner tree: Min cost Steiner tree: Min cost

tree spanning tree spanning SS NP-hard, MST is a 2-NP-hard, MST is a 2-

approx, Current best approx, Current best 1.55-approx (Robins, 1.55-approx (Robins, Zelikovsky ’99)Zelikovsky ’99)

Primal-dual 2-approxPrimal-dual 2-approx (Agrawal, Klein, R. ’91; (Agrawal, Klein, R. ’91; Goemans, Williamson Goemans, Williamson ’92)’92)

Page 35: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

3535

Stochastic Min. Steiner TreeStochastic Min. Steiner Tree

Given a Given a metric metric space of pointsspace of points, , distances distances ccee

Points: Points: possible possible locationslocations of future of future demanddemand

Wlog, simplifying Wlog, simplifying assumption: no 1assumption: no 1stst stage demandstage demand

Page 36: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

3636

Stochastic Min. Steiner TreeStochastic Min. Steiner Tree

Given a metric Given a metric space of points, space of points, distances distances ccee

11stst stage stage: buy : buy edges at costs edges at costs ccee

Page 37: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

3737

Stochastic Min. Steiner TreeStochastic Min. Steiner Tree

Given a metric space Given a metric space of points, distances of points, distances ccee

11stst stage: buy edges stage: buy edges at costs at costs ccee

22ndnd stage stage: Some : Some clients “realized”, buy clients “realized”, buy edges at cost edges at cost σσ.c.cee to to serve them (serve them (σσ > 1 > 1))

Page 38: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

3838

Stochastic Min. Steiner TreeStochastic Min. Steiner Tree

Given a metric space Given a metric space of points, distances of points, distances ccee

11stst stage: buy edges stage: buy edges at costs at costs ccee

22ndnd stage stage: Some : Some clients “realized”, buy clients “realized”, buy edges at cost edges at cost σσ.c.cee to to serve them (serve them (σσ > 1 > 1))

Page 39: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

3939

Stochastic Min. Steiner TreeStochastic Min. Steiner Tree

Given a metric space Given a metric space of points, distances of points, distances ccee

11stst stage: buy edges stage: buy edges at costs at costs ccee

22ndnd stage: Some stage: Some clients “realized”, buy clients “realized”, buy edges at cost edges at cost σσ.c.cee to to serve them (serve them (σσ > 1 > 1))

Minimize exp. costMinimize exp. cost

Page 40: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

4040

Algorithm Boosted-SampleAlgorithm Boosted-Sample

Sample Sample from the distribution of from the distribution of clientsclients σσ times (sampled set times (sampled set SS))

Page 41: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

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Algorithm Boosted-SampleAlgorithm Boosted-Sample

Sample from the distribution of Sample from the distribution of clients clients σσ times (sampled set times (sampled set SS))

Build Build minimum spanning tree minimum spanning tree TT00 on on SS Recall: Minimum spanning tree is a 2-Recall: Minimum spanning tree is a 2-

approximation to Minimum Steiner treeapproximation to Minimum Steiner tree

Page 42: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

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Algorithm Boosted-SampleAlgorithm Boosted-Sample

Sample from the distribution of Sample from the distribution of clients clients σσ times (sampled set times (sampled set SS))

Build minimum spanning tree Build minimum spanning tree TT00 on on SS 22ndnd stage: actual client set realized stage: actual client set realized

((RR))

- - Extend Extend TT00 to span to span RR

Page 43: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

4343

Algorithm Boosted-SampleAlgorithm Boosted-Sample

Sample from the distribution of Sample from the distribution of clients clients σσ times (sampled set times (sampled set SS))

Build minimum spanning tree Build minimum spanning tree TT00 on on SS 22ndnd stage: actual client set realized stage: actual client set realized

((RR))

- Extend - Extend TT00 to span to span RR Theorem: 4-approximation!Theorem: 4-approximation!

Page 44: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

4444

Algorithm: IllustrationAlgorithm: Illustration

Input, with Input, with σσ=3=3

Page 45: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

4545

Algorithm: IllustrationAlgorithm: Illustration

Input, with Input, with σσ=3=3 SampleSample σσ times times

from client from client distributiondistribution

Page 46: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

4646

Algorithm: IllustrationAlgorithm: Illustration

Input, with Input, with σσ=3=3 Sample Sample σσ times times

from client from client distributiondistribution

Page 47: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

4747

Algorithm: IllustrationAlgorithm: Illustration

Input, with Input, with σσ=3=3 SampleSample σσ times times

from client from client distributiondistribution

Page 48: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

4848

Algorithm: IllustrationAlgorithm: Illustration

Input, with Input, with σσ=3=3 Sample Sample σσ times times

from client from client distributiondistribution

Page 49: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

4949

Algorithm: IllustrationAlgorithm: Illustration

Input, with Input, with σσ=3=3 Sample Sample σσ times times

from client from client distributiondistribution

Build MSTBuild MST TT00 on on SS

Page 50: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

5050

Algorithm: IllustrationAlgorithm: Illustration

Input, with Input, with σσ=3=3 Sample Sample σσ times times

from client from client distributiondistribution

Build MST Build MST TT00 on on SS When When actual actual

scenarioscenario ( (RR) is ) is realized …realized …

Page 51: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

5151

Algorithm: IllustrationAlgorithm: Illustration

Input, with Input, with σσ=3=3 Sample Sample σσ times times

from client from client distributiondistribution

Build MST Build MST TT00 on on SS When actual When actual

scenario (scenario (RR) is ) is realized …realized …

ExtendExtend TT00 to span to span RR

Page 52: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

5252

Analysis of 1Analysis of 1stst stage cost stage cost

Let Let X

XX TcpTcOPT )()( **0

Page 53: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

5353

Analysis of 1Analysis of 1stst stage cost stage cost

Let Let ClaimClaim::

X

XX TcpTcOPT )()( **0

OPTTcE .2)]([ 0

Page 54: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

5454

Analysis of 1Analysis of 1stst stage cost stage cost

Let Let Claim:Claim:

Our Our σσ samplessamples: : S={SS={S11, S, S22, …, S, …, Sσσ}}

X

XX TcpTcOPT )()( **0

)}(...)()({2)( ***0 1 SS TcTcTcSMST

OPTTcE .2)]([ 0

Page 55: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

5555

Analysis of 1Analysis of 1stst stage cost stage cost

Let Let Claim:Claim:

Our Our σσ samples: samples: S={SS={S11, S, S22, …, S, …, Sσσ}}

X

XX TcpTcOPT )()( **0

OPTTcE .2)]([ 0

)}(...)()({2)( ***0 1 SS TcTcTcSMST

)]}([...)]([)({2)]([ ***0 1 SS TcETcETcSMSTE

Page 56: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

5656

Analysis of 1Analysis of 1stst stage cost stage cost

Let Let Claim:Claim:

Our Our σσ samples: samples: S={SS={S11, S, S22, …, S, …, Sσσ}}

X

XX TcpTcOPT )()( **0

)}(...)()({2)( ***0 1 SS TcTcTcSMST

)]}([...)]([)({2)]([ ***0 1 SS TcETcETcSMSTE

)]}([)({2 **0 XX TcETc

OPTTcE .2)]([ 0

Page 57: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

5757

Analysis of 2Analysis of 2ndnd stage cost stage cost

Intuition:Intuition: 11stst stage: stage: σσ samples at cost samples at cost ccee

22ndnd stage: stage: 11 sample at cost sample at cost σσ.c.cee

Page 58: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

5858

Analysis of 2Analysis of 2ndnd stage cost stage cost

Intuition:Intuition: 11stst stage: stage: σσ samples at cost samples at cost ccee

22ndnd stage: stage: 11 sample at cost sample at cost σσ.c.cee

In expectation,In expectation, 22ndnd stage cost ≤ 1 stage cost ≤ 1stst stage cost stage cost

Page 59: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

5959

Analysis of 2Analysis of 2ndnd stage cost stage cost

Intuition:Intuition: 11stst stage: stage: σσ samples at cost samples at cost ccee

22ndnd stage: stage: 11 sample at cost sample at cost σσ.c.cee

In expectation,In expectation, 22ndnd stage cost ≤ 1 stage cost ≤ 1stst stage cost stage cost

But we’ve But we’ve already bounded 1already bounded 1stst stage stage cost!cost!

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Analysis of 2Analysis of 2ndnd stage cost stage cost

Claim: Claim: E[E[σσc(Tc(TRR)] ≤ )] ≤ E[c(TE[c(T00)])]

Proof using an Proof using an auxiliary auxiliary structurestructure

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Analysis of 2Analysis of 2ndnd stage cost stage cost

Claim: Claim: E[E[σσc(Tc(TRR)] ≤ )] ≤ E[c(TE[c(T00)])]

Let Let TTRSRS be an be an MST MST on on R U R U SS

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6262

Analysis of 2Analysis of 2ndnd stage cost stage cost

Claim: Claim: E[E[σσc(Tc(TRR)] ≤ )] ≤ E[c(TE[c(T00)])]

Let Let TTRSRS be an MST on be an MST on R U R U SS

Associate each node Associate each node v v єє TTRSRS with its with its parent edgeparent edge pt(v)pt(v); ; c(Tc(TRSRS)=c(pt(R)) + )=c(pt(R)) + c(pt(S))c(pt(S))

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Analysis of 2Analysis of 2ndnd stage cost stage cost

Claim: Claim: E[E[σσc(Tc(TRR)] ≤ E[c(T)] ≤ E[c(T00)])] Let Let TTRSRS be an MST on be an MST on R U SR U S Associate each node Associate each node v v єє T TRSRS

with its parent edge with its parent edge pt(v); pt(v); c(Tc(TRSRS)=c(pt(R)) + c(pt(S)))=c(pt(R)) + c(pt(S))

c(Tc(TRR) ≤ c(pt(R))) ≤ c(pt(R)), since , since TTRR was the was the cheapest possible cheapest possible way to connectway to connect R R to to TT00

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6464

Analysis of 2Analysis of 2ndnd stage cost stage cost

Claim: Claim: E[E[σσc(Tc(TRR)] ≤ E[c(T)] ≤ E[c(T00)])] Let Let TTRSRS be an MST on be an MST on R U SR U S Associate each node Associate each node v v єє T TRSRS

with its parent edge with its parent edge pt(v); pt(v); c(Tc(TRSRS)=c(pt(R)) + c(pt(S)))=c(pt(R)) + c(pt(S))

c(Tc(TRR) ≤ c(pt(R))) ≤ c(pt(R))

E[c(pt(R))] ≤ E[c(pt(S))]/E[c(pt(R))] ≤ E[c(pt(S))]/σσ,,

since since RR is is 11 sample and sample and SS is is σσ samples from samples from same same processprocess

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6565

Analysis of 2Analysis of 2ndnd stage cost stage cost

Claim: Claim: E[E[σσc(Tc(TRR)] ≤ E[c(T)] ≤ E[c(T00)])] Let Let TTRSRS be an MST on be an MST on R U SR U S Associate each node Associate each node v v єє

TTRSRS with its parent edge with its parent edge pt(v); c(Tpt(v); c(TRSRS)=c(pt(R)) + )=c(pt(R)) + c(pt(S))c(pt(S))

c(Tc(TRR) ≤ c(pt(R))) ≤ c(pt(R)) E[c(pt(R))] ≤ E[c(pt(S))]/E[c(pt(R))] ≤ E[c(pt(S))]/σσ

c(pt(S)) ≤ c(Tc(pt(S)) ≤ c(T00)),,since since pt(S) U pt(R) pt(S) U pt(R) is a is a MST MST while adding while adding pt(R)pt(R) to to TT00 spans spans R U S R U S

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Analysis of 2Analysis of 2ndnd stage cost stage cost

Claim: Claim: E[E[σσc(Tc(TRR)] ≤ E[c(T)] ≤ E[c(T00)])] Let Let TTRSRS be an MST on be an MST on R U SR U S Associate each node Associate each node v v єє

TTRSRS with its parent edge with its parent edge pt(v); c(Tpt(v); c(TRSRS)=c(pt(R)) + )=c(pt(R)) + c(pt(S))c(pt(S))

c(Tc(TRR) ≤ c(pt(R))) ≤ c(pt(R)) E[c(pt(R))] ≤ E[c(pt(S))]/E[c(pt(R))] ≤ E[c(pt(S))]/σσ c(pt(S)) ≤ c(Tc(pt(S)) ≤ c(T00))

Chain inequalities and Chain inequalities and claim followsclaim follows

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RecapRecap

Algorithm for Stochastic Steiner Tree:Algorithm for Stochastic Steiner Tree: 11stst stage: stage: SampleSample σσ times, build MST times, build MST 22ndnd stage: stage: Extend Extend MST to realized clientsMST to realized clients

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RecapRecap

Algorithm for Stochastic Steiner Tree:Algorithm for Stochastic Steiner Tree: 11stst stage: Sample stage: Sample σσ times, build MST times, build MST 22ndnd stage: Extend MST to realized clients stage: Extend MST to realized clients

TheoremTheorem: Algorithm BOOST-AND-: Algorithm BOOST-AND-SAMPLE is a 4-approximation to SAMPLE is a 4-approximation to Stochastic Steiner TreeStochastic Steiner Tree

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RecapRecap

Algorithm for Stochastic MST:Algorithm for Stochastic MST: 11stst stage: Sample stage: Sample σσ times, build MST times, build MST 22ndnd stage: Extend MST to realized clients stage: Extend MST to realized clients

Theorem: Algorithm BOOST-AND-SAMPLE is a Theorem: Algorithm BOOST-AND-SAMPLE is a 4-approximation to Stochastic Steiner Tree4-approximation to Stochastic Steiner Tree

Shortcomings: Shortcomings: Specific problem, in a specific modelSpecific problem, in a specific model Cannot adapt to scenario model with non-Cannot adapt to scenario model with non-

correlated cost changes across scenarioscorrelated cost changes across scenarios

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Coping with shortcomingsCoping with shortcomings

Specific problem, in a specific modelSpecific problem, in a specific model Boosted Sampling works for more general covering problems Boosted Sampling works for more general covering problems

with subadditivity - Solves Facility location, vertex coverwith subadditivity - Solves Facility location, vertex cover

Skip general model (details in STOC 04 paper)Skip general model (details in STOC 04 paper)

Cannot adapt to scenario model with scenario-Cannot adapt to scenario model with scenario-dependent cost inflationsdependent cost inflations A combination of LP-rounding and primal-dual methods A combination of LP-rounding and primal-dual methods

solves the scenario model with scenario-dependent cost solves the scenario model with scenario-dependent cost inflations; Also handles risk-bounds on more general network inflations; Also handles risk-bounds on more general network design.design.Skip scenario model (details in FOCS 04 paper)Skip scenario model (details in FOCS 04 paper)

Skip bothSkip both

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OutlineOutline

Motivation: The cable company Motivation: The cable company problemproblem

Model and literature reviewModel and literature review Solution to the cable company Solution to the cable company

problemproblem General covering problemGeneral covering problem Scenario dependent cost model Scenario dependent cost model

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General ModelGeneral Model

U U : universe of potential clients (e.g., : universe of potential clients (e.g., terminals)terminals)

X X : elements which provide service, : elements which provide service, with element costs with element costs ccx x (e.g., edges)(e.g., edges)

Given Given S S UU, set of feasible sol’ns is , set of feasible sol’ns is Sols(Sols(S S ) ) 22XX

Deterministic problem: Given Deterministic problem: Given SS, find , find minimum cost minimum cost FF Sols( Sols(S S ))

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Model: detailsModel: details

Element costs are Element costs are ccxx in first stage and in first stage and σσ.c.cxx in second stage in second stage

In second stage, client set In second stage, client set SS UU is is realized with probability realized with probability p(S)p(S)

Objective: Compute Objective: Compute FF0 0 and and FFS S to to minimizeminimize

c(Fc(F00) + E[) + E[σσ c(F c(FSS)])]

where where FF00 F FSS Sols( Sols(S S ) for all ) for all SS

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Sampling access modelSampling access model

Second stage: Client set Second stage: Client set S S appears appears with probability with probability p(S)p(S)

We only require sampling access:We only require sampling access: Oracle, when queried, gives us a sample Oracle, when queried, gives us a sample

scenario scenario DD Identically distributed to actual second Identically distributed to actual second

stagestage

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Main result: PreviewMain result: Preview

Given stochastic optimization problem Given stochastic optimization problem with cost inflation factor with cost inflation factor σσ :: Generate Generate σσ samples: samples: DD11, D, D22, …, D, …, Dσσ

Use deterministic approximation algorithm to Use deterministic approximation algorithm to compute compute FF00 Sols( Sols(DDi i ))

When actual second stage When actual second stage S S is realized, is realized, augment by selecting augment by selecting FFSS

Theorem: Good approximation for Theorem: Good approximation for stochastic problem!stochastic problem!

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Requirement: Sub-additivityRequirement: Sub-additivity

If If S S and and S’S’ are legal sets of clients, are legal sets of clients, then:then: SS S’S’ is also a legal client set is also a legal client set For any For any FF Sols( Sols(S S ) and ) and F’F’ Sols( Sols(S’ S’ ), ),

we also have we also have F F F’F’ Sols( Sols(SS S’ S’ ) )

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Requirement: Requirement: ApproximationApproximation

There is an There is an -approximation -approximation algorithm for deterministic problemalgorithm for deterministic problem Given any Given any SS UU, can find , can find FF Sols( Sols(S S ) in ) in

polynomial time such that:polynomial time such that:

c(F)c(F) .min {.min {c(F’)c(F’): : F’F’ Sols( Sols(S S )})}

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Crucial ingredient: Cost Crucial ingredient: Cost sharesshares

Recall Stochastic Steiner Tree:Recall Stochastic Steiner Tree: Bounding 2Bounding 2ndnd stage cost required allocating the stage cost required allocating the

cost of an MST to the client nodes, and summing cost of an MST to the client nodes, and summing up carefully (auxiliary structure)up carefully (auxiliary structure)

Cost sharing functionCost sharing function: way of distributing : way of distributing solution cost to clientssolution cost to clients

Originated in game theory [Young, ’94], Originated in game theory [Young, ’94], adapted to approximation algorithms adapted to approximation algorithms [Gupta, Kumar, Pal, Roughgarden FOCS ’03][Gupta, Kumar, Pal, Roughgarden FOCS ’03]

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Requirement: Cost-sharingRequirement: Cost-sharing

ξξ : : 22U U x x UU R is a R is a ββ -strict cost sharing -strict cost sharing function for function for -approximation A if:-approximation A if: ξξ(S,j)(S,j) > 0 only if > 0 only if jj SS jjS S ξξ(S,j) (S,j) c c (OPT((OPT(S S )))) If If S’S’ = = SS TT, A(, A(S S ) is an ) is an -approx. for -approx. for SS, ,

and Aug(and Aug(SS,,T T ) provides a solution for ) provides a solution for augmenting A(augmenting A(S S ) to also serve ) to also serve TT, then, then

jjT T ξξ(S’,j) (S’,j) ( (1/1/ββ ) ) c c (Aug((Aug(S,T S,T ))))

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Main theorem: FormalMain theorem: Formal

Given a sub-additive problem with Given a sub-additive problem with --approximation algorithm A and approximation algorithm A and ββ-strict cost -strict cost sharing function, the following is an (sharing function, the following is an (++ββ )-)-approximation algorithm for stochastic approximation algorithm for stochastic variant:variant: Generate Generate σσ samples: samples: DD11, D, D22, …, D, …, Dσσ

First stage: Use algorithm A to compute First stage: Use algorithm A to compute FF00 as an as an -approximation for -approximation for DDii

Second stage: When actual set Second stage: When actual set S S is realized, use is realized, use algorithm Aug(algorithm Aug( DDi i , , S S ) to compute ) to compute FFSS

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First-stage costFirst-stage cost

Samples Samples DDi i , Algo A generates , Algo A generates FF00 Sols( Sols( DDi i )) Define optimum: Define optimum: ZZ** = = c(Fc(F00

**) + ) + SS p(S). p(S).σσ.c(F.c(FSS**))

By sub-additivity, By sub-additivity,

FF00** FFD1D1

** … … FFDDσσ** Sols( Sols( DDi i ))

Since A is Since A is -approximation,-approximation,

c(Fc(F0 0 )/)/ c(Fc(F00**) + ) + ii c(F c(FDiDi

**)) E[E[c(Fc(F0 0 ))]/]/ c(Fc(F00

**) + ) + ii E[ E[c(Fc(FSS**))] ]

c(Fc(F00**) + ) + σσ SS p(S) c(F p(S) c(FSS

**) = Z) = Z**

Therefore, first-stage cost E[Therefore, first-stage cost E[c(Fc(F00))] ] .Z.Z**

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Second-stage costSecond-stage cost

DDi i : samples, : samples, S S : actual 2: actual 2ndnd stage, define stage, define S’S’ = = S S DDii

c(Fc(FSS) ) ββ..ξξ(S’,S)(S’,S), by cost-sharing function defn., by cost-sharing function defn. ξξ(S’ ,D(S’ ,D11) + … + ) + … + ξξ(S’ ,D(S’ ,Dσσ) + ) + ξξ(S’ ,S) (S’ ,S) c c (OPT((OPT(S’ S’ )))) S’S’ has has σσ+1+1 client sets, identically distributed: client sets, identically distributed:

E[E[ξξ(S’ ,S)(S’ ,S)] ] E[ E[c c (OPT((OPT(S’ S’ ))] / ))] / ((σσ+1)+1) c c (OPT((OPT(S’ S’ )) )) c(Fc(F00

**) + c(F) + c(FD1D1**) + … + c(F) + … + c(FDDσσ

**) + c(F) + c(FSS**)),,

by sub-additivityby sub-additivity E[E[c c (OPT((OPT(S’ S’ ))] ))] c(Fc(F00

**) + () + (σσ+1)+1) E[ E[c(Fc(Fss))] ] ((σσ+1)Z+1)Z**//σσ E[E[σσ.c(F.c(FSS))] ] ββ.Z.Z**, bounding second-stage cost, bounding second-stage cost

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OutlineOutline

Motivation: The cable company problemMotivation: The cable company problem Model and literature reviewModel and literature review Solution to the cable company problemSolution to the cable company problem General covering problemGeneral covering problem Scenario dependent cost modelScenario dependent cost model

Page 84: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

8484

Stochastic Steiner TreeStochastic Steiner Tree

First stage: First stage: GG, , rr given given 22ndnd stage: one of stage: one of mm

scenariosscenarios occurs: occurs: TerminalsTerminals SSkk

ProbabilityProbability ppkk

Edge Edge cost inflationcost inflation factor factor σσkk

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8585

Stochastic Steiner TreeStochastic Steiner Tree

First stage: First stage: GG, , rr given given 22ndnd stage: one of stage: one of mm

scenarios occurs:scenarios occurs: Terminals Terminals SSkk

Probability Probability ppkk

Edge cost inflation Edge cost inflation factor factor σσkk

ObjectiveObjective: 1: 1stst stage stage tree tree TT00, 2, 2ndnd stage trees stage trees TTkk s.t. s.t. TT00TTkk span span SSkk

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8686

Stochastic Steiner TreeStochastic Steiner Tree

First stage: First stage: GG, , rr given given 22ndnd stage: one of stage: one of mm

scenarios occurs:scenarios occurs: Terminals Terminals SSkk

Probability Probability ppkk

Edge cost inflation Edge cost inflation factor factor σσkk

Objective: Objective: 11stst stage stage tree tree TT00, 2, 2ndnd stage trees stage trees TTkk s.t. s.t. TT00TTkk span span SSkk

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8787

Stochastic Steiner TreeStochastic Steiner Tree

First stage: First stage: GG, , rr given given 22ndnd stage: one of stage: one of mm

scenarios occursscenarios occurs:: Terminals Terminals SSkk

Probability Probability ppkk

Edge cost inflation Edge cost inflation factor factor σσkk

Objective: 1Objective: 1stst stage stage tree tree TT00, 2, 2ndnd stage trees stage trees TTkk s.t. s.t. TT00TTkk span span SSkk

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8888

Stochastic Steiner TreeStochastic Steiner Tree

First stage: First stage: GG, , rr given given 22ndnd stage: one of stage: one of mm

scenarios occurs:scenarios occurs: Terminals Terminals SSkk

Probability Probability ppkk

Edge cost inflation Edge cost inflation factor factor σσkk

Objective: 1Objective: 1stst stage stage tree tree TT00, , 22ndnd stage trees stage trees TTkk s.t. s.t. TT00TTkk span span SSkk

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8989

Stochastic Steiner TreeStochastic Steiner Tree

First stage: First stage: GG, , rr given given 22ndnd stage: one of stage: one of mm

scenarios occurs:scenarios occurs: Terminals Terminals SSkk

Probability Probability ppkk

Edge cost inflation factor Edge cost inflation factor σσkk

Objective: 1Objective: 1stst stage tree stage tree TT00, 2, 2ndnd stage trees stage trees TTkk s.t. s.t. TT00TTkk span span SSkk

Minimize Minimize c(Tc(T00)+E[c(T’)])+E[c(T’)] Skip AlgorithmSkip Algorithm

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Tree solutionsTree solutions

Example with 4 Example with 4 scenarios and scenarios and σσ=2=2

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Tree solutionsTree solutions

Example with 4 Example with 4 scenarios and scenarios and σσ=2=2

Optimal solution may Optimal solution may have have lots of lots of componentscomponents!!

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Tree solutionsTree solutions

Example with 4 Example with 4 scenarios and scenarios and σσ=2=2

Optimal solution may Optimal solution may have lots of have lots of components!components!

LemmaLemma: There exists a : There exists a solution where solution where 11stst stage stage is a treeis a tree and overall and overall cost is no more than 3 cost is no more than 3 times the optimal costtimes the optimal cost

Restrict to tree solutionsRestrict to tree solutions

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9393

IP formulationIP formulation

Tree solutionTree solution: From any (2: From any (2ndnd-stage) -stage) terminal, path to root consists of terminal, path to root consists of exactly two parts: strictly 2exactly two parts: strictly 2ndnd-stage, -stage, followed by strictly 1followed by strictly 1stst-stage-stage

IP: IP: Install edges to support unit flowInstall edges to support unit flow along such paths from each terminal along such paths from each terminal to rootto root

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9494

IP formulationIP formulation

0)(

0)()(

0))()(())()((

1))()((

)()(min

)(

0

)(

0)(

0

)(

0)(

01

0

ke

ke

vee

vee

ve

kee

ve

kee

te

kee

Ee

m

k Ee

kekke

xtr

trtr

trtrtrtr

trtr

xecpxec

xek: edge e installed in scenario k;

rek(t): flow on edge e of type k from terminal t ;

for k = 0 (1st stage) and i=1,2,…,m (2nd stage)

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9595

IP formulationIP formulation

0)(

0)()(

0))()(())()((

1))()((

)()(min

)(

0

)(

0)(

0

)(

0)(

01

0

ke

ke

vee

vee

ve

kee

ve

kee

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kee

Ee

m

k Ee

kekke

xtr

trtr

trtrtrtr

trtr

xecpxec

Objective: minimize expected cost

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9696

IP formulationIP formulation

0)(

0)()(

0))()(())()((

1))()((

)()(min

)(

0

)(

0)(

0

)(

0)(

01

0

ke

ke

vee

vee

ve

kee

ve

kee

te

kee

Ee

m

k Ee

kekke

xtr

trtr

trtrtrtr

trtr

xecpxec

Unit out-flow from each terminal

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9797

IP formulationIP formulation

0)(

0)()(

0))()(())()((

1))()((

)()(min

)(

0

)(

0)(

0

)(

0)(

01

0

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ke

vee

vee

ve

kee

ve

kee

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kekke

xtr

trtr

trtrtrtr

trtr

xecpxec

Flow conservation at all internal nodes (v ≠ t , r )

Page 98: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

9898

IP formulationIP formulation

0)(

0)()(

0))()(())()((

1))()((

)()(min

)(

0

)(

0)(

0

)(

0)(

01

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kekke

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trtr

trtrtrtr

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xecpxec

Flow monotonicity: enforces “First-stage must be a tree”

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9999

IP formulationIP formulation

0)(

0)()(

0))()(())()((

1))()((

)()(min

)(

0

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0)(

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0)(

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trtrtrtr

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xecpxec

Flow support: If an edge has flow, it must be accounted for in the objective function

Page 100: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

100100

IP formulationIP formulation

}1,0{),(

0),(

0)(

0)()(

0))()(())()((

1))()((

)()(min

)(

0

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Algorithm overviewAlgorithm overview

((x,rx,r) ) Optimal solution to LPOptimal solution to LP relaxationrelaxation

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Algorithm overviewAlgorithm overview

((x,rx,r) ) Optimal solution to LP Optimal solution to LP relaxationrelaxation

11stst stage solution stage solution: : Obtain a new graph Obtain a new graph G’G’ where where 2x2x00 forms forms

a fractional Steiner treea fractional Steiner tree Round using primal-dual algorithm; this Round using primal-dual algorithm; this

is is TT00

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Algorithm overviewAlgorithm overview

((x,rx,r) ) Optimal solution to LP relaxation Optimal solution to LP relaxation 11stst stage solution: stage solution:

Obtain a new graph Obtain a new graph G’G’ where where 2x2x00 forms a forms a fractional Steiner treefractional Steiner tree

Round using primal-dual algorithm; this is Round using primal-dual algorithm; this is TT00 22ndnd stage solution stage solution::

Examine remaining terminals in each scenarioExamine remaining terminals in each scenario Use modified primal-dual method to obtain Use modified primal-dual method to obtain TTk k

SkipSkip Analysis Analysis

Page 104: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

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First stageFirst stage

Examine Examine fractional fractional pathspaths for each for each terminalterminal

Page 105: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

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First stageFirst stage

Examine fractional Examine fractional paths for each paths for each terminalterminal

Critical radiusCritical radius: Flow : Flow “transitions” from 2“transitions” from 2ndnd--stage to 1stage to 1stst-stage-stage

Page 106: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

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First stageFirst stage

Examine fractional Examine fractional paths for each paths for each terminalterminal

Critical radius: Flow Critical radius: Flow “transitions” from 2“transitions” from 2ndnd--stage to 1stage to 1stst-stage-stage

Construct Construct critical radii critical radii for all terminalsfor all terminals

Page 107: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

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First stageFirst stage

Critical radius: Critical radius: Fractional flow Fractional flow “transitions” from 2“transitions” from 2ndnd--stage to 1stage to 1stst-stage-stage

Construct Construct twice the twice the critical radiicritical radii for all for all terminalsterminals

Page 108: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

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First stageFirst stage

Critical radius: Critical radius: Fractional flow Fractional flow “transitions” from 2“transitions” from 2ndnd--stage to 1stage to 1stst-stage-stage

Construct twice the Construct twice the c.r.c.r. for all terminals for all terminals

Examine in increasing Examine in increasing order of order of c.r.c.r.

RR00 independent set independent set based on based on 2 2 ××c.r.c.r.

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First stageFirst stage

Critical radius: Critical radius: Fractional flow Fractional flow “transitions” from 2“transitions” from 2ndnd--stage to 1stage to 1stst-stage-stage

Construct twice the Construct twice the c.r.c.r. for all terminals for all terminals

Examine in increasing Examine in increasing order of order of c.r.c.r.

RR00 independent set independent set based on based on 2 2 ××c.r.c.r.

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First stageFirst stage

Critical radius: Critical radius: Fractional flow Fractional flow “transitions” from 2“transitions” from 2ndnd--stage to 1stage to 1stst-stage-stage

Construct twice the Construct twice the c.r.c.r. for all terminals for all terminals

Examine in increasing Examine in increasing order of order of c.r.c.r.

RR00 independent set independent set based on based on 2 2 ××c.r.c.r.

Page 111: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

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First stageFirst stage

Critical radius: Critical radius: Fractional flow Fractional flow “transitions” from 2“transitions” from 2ndnd--stage to 1stage to 1stst-stage-stage

Construct twice the Construct twice the c.r.c.r. for all terminalsfor all terminals

Examine in increasing Examine in increasing order of order of c.r.c.r.

RR00 independent set independent set based on based on 2 2 ××c.r.c.r.

TT00 Steiner tree on Steiner tree on RR00

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First stage analysisFirst stage analysis

Critical radius: Critical radius: Fractional flow Fractional flow “transitions” from 2“transitions” from 2ndnd--stage to 1stage to 1stst-stage-stage

RR00 independent set independent set based on based on 2 2 ××c.r.c.r.

TT00 Steiner tree on Steiner tree on RR00

G’G’ Contract Contract c.r.c.r. balls balls around vertices in around vertices in RR00

2x2x00 is feasible is feasible fractional fractional Steiner tree for Steiner tree for RR00 in in G’G’

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First stage analysisFirst stage analysis

RR00 independent set independent set based on based on 2 2 ××c.r.c.r.

TT00 Steiner tree on Steiner tree on RR00

G’G’ Contract Contract c.r.c.r. balls balls around vertices in around vertices in RR00

2x2x00 is feasible fractional is feasible fractional Steiner tree for Steiner tree for RR00 in in G’G’

Extension from vertex to Extension from vertex to c.r.c.r. charged to segment charged to segment from from c.r.c.r. to to 2 2 ××c.r.c.r. (disjoint from others)(disjoint from others)

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Second stageSecond stage

TT00 1 1stst stage tree stage tree Consider scenario Consider scenario kk

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Second stageSecond stage

TT00 1 1stst stage tree stage tree Consider scenario Consider scenario kk Idea: Run Steiner tree Idea: Run Steiner tree

primal-dual on primal-dual on terminals, stopping terminals, stopping moat moat MM when: when:

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Second stageSecond stage

TT00 1 1stst stage tree stage tree Consider scenario Consider scenario kk Idea: Run Steiner tree Idea: Run Steiner tree

primal-dual on primal-dual on terminals, stopping terminals, stopping moat moat MM when: when: MM hits hits TT00

MM hits a stopped moat hits a stopped moat For every terminal in For every terminal in MM, ,

less than ½ flow leaving less than ½ flow leaving MM is 2 is 2ndnd-stage -stage

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Second stageSecond stage

TT00 1 1stst stage tree stage tree Consider scenario Consider scenario kk Idea: Run Steiner tree Idea: Run Steiner tree

primal-dual on primal-dual on terminals, stopping terminals, stopping moat moat MM when: when: MM hits hits TT00

MM hits a stopped moat hits a stopped moat For every terminal in For every terminal in MM, ,

less than ½ flow leaving less than ½ flow leaving MM is 2 is 2ndnd-stage -stage

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Second stageSecond stage

TT00 1 1stst stage tree stage tree Consider scenario Consider scenario kk Idea: Run Steiner tree Idea: Run Steiner tree

primal-dual on primal-dual on terminals, stopping terminals, stopping moat moat MM when: when: MM hits hits TT00

MM hits a stopped moat hits a stopped moat For every terminal in For every terminal in MM, ,

less than ½ flow leaving less than ½ flow leaving MM is 2 is 2ndnd-stage -stage

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Second stageSecond stage

TT00 1 1stst stage tree stage tree Consider scenario Consider scenario kk Idea: Run Steiner tree Idea: Run Steiner tree

primal-dual on primal-dual on terminals, stopping terminals, stopping moat moat MM when: when: MM hits hits TT00

MM hits a stopped moat hits a stopped moat For every terminal in For every terminal in MM, ,

less than ½ flow leaving less than ½ flow leaving MM is 2 is 2ndnd-stage -stage

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Second stageSecond stage

TT00 1 1stst stage tree stage tree Consider scenario Consider scenario kk Idea: Run Steiner tree Idea: Run Steiner tree

primal-dual on primal-dual on terminals, stopping terminals, stopping moat moat MM when: when: MM hits hits TT00

MM hits a stopped moat hits a stopped moat For every terminal in For every terminal in MM, ,

less than ½ flow leaving less than ½ flow leaving MM is 2 is 2ndnd-stage -stage

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Second stageSecond stage

Idea: Run Steiner tree Idea: Run Steiner tree primal-dual on primal-dual on terminals, stopping terminals, stopping moat moat MM when: when: MM hits hits TT00

MM hits a stopped moat hits a stopped moat For every terminal in For every terminal in MM, ,

less than ½ flow leaving less than ½ flow leaving MM is 2 is 2ndnd-stage -stage

If If MM hits hits TT00, add edge , add edge from from ttMM to to vvRR00

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Second stageSecond stage

Idea: Run Steiner tree Idea: Run Steiner tree primal-dual on primal-dual on terminals, stopping terminals, stopping moat moat MM when: when: MM hits hits TT00

MM hits a stopped moat hits a stopped moat For every terminal in For every terminal in MM, ,

less than ½ flow leaving less than ½ flow leaving MM is 2is 2ndnd-stage -stage

If If MM hits hits M’M’, connect , connect ttMM with with t’t’M’M’ as in Steiner as in Steiner tree primal-dualtree primal-dual

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Second stageSecond stage

Idea: Run Steiner tree Idea: Run Steiner tree primal-dual on terminals, primal-dual on terminals, stopping moat stopping moat MM when: when: MM hits hits TT00

MM hits a stopped moat hits a stopped moat For every terminal in For every terminal in MM, ,

less than ½ flow leaving less than ½ flow leaving MM is 2is 2ndnd-stage -stage

There exists There exists ttMM and and vvRR0 0 s.t. s.t. vv within within 4 4 ××c.r.c.r. of of t t ; connect ; connect t t to to vv

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Second stage analysisSecond stage analysis

Primal-dual accounts Primal-dual accounts for edges inside moatsfor edges inside moats

Page 125: Approximation Algorithms for Stochastic Combinatorial Optimization R. Ravi Carnegie Mellon University Joint work with: Kedar Dhamdhere, CMU Kedar Dhamdhere,

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Second stage analysisSecond stage analysis

Primal-dual accounts Primal-dual accounts for edges inside moatsfor edges inside moats

Connector edgesConnector edges paid paid by carefully by carefully accounting:accounting: Primal-dual boundPrimal-dual bound For every terminal For every terminal tt, ,

there is there is vvRR00 within within 4 4 ××c.r.c.r. of of tt

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SST: main resultSST: main result

24-approximation24-approximation for for Stochastic Steiner Tree Stochastic Steiner Tree (Improvement to 16-(Improvement to 16-approx possible)approx possible)

Method:Method: Primal-dual Primal-dual overlaid on LP solutionoverlaid on LP solution

Extensions to more Extensions to more general network design general network design with routing costswith routing costs

Per-scenario risk-bounds Per-scenario risk-bounds incorporated and incorporated and roundedrounded

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Main Techniques in other Main Techniques in other resultsresults

Stochastic Facility LocationStochastic Facility Location – Rounding natural LP – Rounding natural LP formulation using filter-and-round (Lin-Vitter, Shmoys-formulation using filter-and-round (Lin-Vitter, Shmoys-Tardos-Aardal) carefully [Details in IPCO ’04]Tardos-Aardal) carefully [Details in IPCO ’04]

Stochastic Minimum Spanning TreeStochastic Minimum Spanning Tree – Both scenario – Both scenario and black-box models - Randomized rounding of and black-box models - Randomized rounding of natural LP formulation gives nearly best possible natural LP formulation gives nearly best possible O(log [No. of vertices] + log [max cost/min cost of an O(log [No. of vertices] + log [max cost/min cost of an edge across scenarios]) approximation result [Details edge across scenarios]) approximation result [Details in IPCO ’05]in IPCO ’05]

Multi-stage general covering problems Multi-stage general covering problems – Boosted – Boosted sampling with rejection based on ratio of scenario’s sampling with rejection based on ratio of scenario’s inflation to maximum possible works [manuscript]inflation to maximum possible works [manuscript]

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SummarySummary

Natural boosted sampling algorithm works for a Natural boosted sampling algorithm works for a broad class of stochastic problems in black-box broad class of stochastic problems in black-box model model

Boosted sampling with rejection extends to multi-Boosted sampling with rejection extends to multi-stage covering problems in the black-box modelstage covering problems in the black-box model

Existing techniques can be cleverly adapted for Existing techniques can be cleverly adapted for the scenario model (E.g., LP-rounding for Facility the scenario model (E.g., LP-rounding for Facility location, primal-dual for Vertex Covers, location, primal-dual for Vertex Covers, combination of both for Steiner trees)combination of both for Steiner trees)

Randomized rounding of LP formulations works Randomized rounding of LP formulations works for black-box formulation of spanning treesfor black-box formulation of spanning trees