On Stochastic Minimum Spanning Trees

Kedar DhamdhereComputer Science Department

Joint work with: Mohit Singh, R. Ravi (IPCO 05)

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Outline

• Stochastic Optimization Model• Related Work• Algorithm for Stochastic MST• Conclusion

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Stochastic optimization

• Classical optimization assumes deterministic inputs

• Real world data has uncertainties• [Dantzig ‘55, Beale ‘61] Modeling data

uncertainty as probability distribution over inputs

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Common framework

[Birge, Louveaux 97] Two-stage stochastic opt. with recourse

• Two stages of decision making• Probability dist. governing second stage data

and costs• Solution can always be made feasible in

second stage

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Common framework

[Birge, Louveaux 97] Two-stage stochastic opt. with recourse

• Two stages of decision making• Probability dist. governing second stage data

and costs• Solution can always be made feasible in

second stage

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Common framework

[Birge, Louveaux 97] Two-stage stochastic opt. with recourse

• Two stages of decision making• Probability dist. governing second stage data

and costs• Solution can always be made feasible in

second stage

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Stochastic MST

Today Tomorrow

Prob = 1/4

Prob = 1/2

Prob = 1/4

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Stochastic MST

Today’s cost = 2 Tomorrow’s E[cost] = 1

Prob = 1/4

Prob = 1/2

Prob = 1/4

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The goal

• Approximation algorithm under the scenario model

• NP-hardness • Probability distribution given as a set of

scenarios

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The goal

• Approximation algorithm under the scenario model

• NP-hardness • Probability distribution given as a set of

scenarios

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Related work

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

• Approximation algorithms: Polynomial Scenarios model, several problems

using LP rounding, incl. Vertex Cover, Facility Location, Shortest paths [Ravi, Sinha, IPCO ’04]

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Related work

• Vertex cover and Steiner trees in restricted models studied by [Immorlica, Karger, Minkoff, Mirrokni SODA ’04]

• “Black box” model: A general technique of sampling the future scenarios a few times and constructing a first stage solutions for the samples [Gupta et al 04]

• Rounding for stochastic Set Cover, FPRAS for #P hard Stochastic Set Cover LPs [Shmoys, Swamy FOCS ’04]– 2-approximation for stochastic covering problem given

approximation for the deterministic problem

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Our results: approximation algorithm

• Theorem: There is an O(log nk)-approximation algorithm for the stochastic MST problem

• Hardness: [Flaxman et al 05, Gupta] Stochastic MST is min{log n, log k}-hard to

approximate unless P = NP

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LP formulation

min e c0e x0

e+ i pi (e cie xi

e)

s.t.

e 2 S x0e+ xi

e ¸ 1 8 S ½ V, 1· i· k

xie ¸ 0 8 e 2 E, 0· i· k

Each cut must be covered either in the first

stage or in each scenario of the second stage

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Algorithm: randomized rounding

• Solve the LP formulation– fractional solution: x0

e, xie

• For O(log nk) rounds– Include an edge independent of others in the first

stage solution with probability x0e

– Include an edge independent of others in the ith scenario with probability xi

e

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Example

Today Tomorrow

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Example: round 1

Today Tomorrow

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Example: round 1

Today Tomorrow

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Example: round 2

Today Tomorrow

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Proof idea

• Lemma: Cost paid in each round is at most OPT

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Proof idea

• Lemma: Cost paid in each round is at most OPT

• Lemma: In each round, with probability 1/2, the number of connected components in a scenario decrease by 9/10– At least one edge leaving a component is included

with prob 0.63

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Proof idea

• Lemma: Cost paid in each round is at most OPT

• Lemma: In each round, with probability 1/2, the number of connected components in a scenario decrease by 9/10– At least one edge leaving a component is included

with prob 0.63

• After O(log nk) “successful” rounds, only 1 connected component left in each scanario w.h.p.

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Other models for second stage costs

• Sampling Access: “Black box” available which generates a sample of 2nd stage data

O(log n)-approximation in time poly(n,)– : max ratio by which cost of any edge changes– Sample poly(n,) scenarios from “black box”

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Other models for second stage costs

• Independent costs: second stage cost 2u.a.r [0,1]– Threshold heuristic with performance guarantee OPT + (3)/4

• [Frieze 85] Single stage costs 2u.a.r [0,1];

MST has cost (3)

• [Flaxman et al. 05] Both stage costs 2u.a.r [0,1]; Thresholding heuristic gives cost · (3) – 1/2

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Conclusions

• Tight approximation algorithm for stochastic MST based on randomized rounding

• Extensions to other models for uncertainty in data