On Stochastic Minimum Spanning Trees

Kedar DhamdhereComputer Science Department

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

2Computer Science Department

Kedar Dhamdhere

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

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Kedar Dhamdhere

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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Kedar Dhamdhere

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

5Computer Science Department

Kedar Dhamdhere

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

6Computer Science Department

Kedar Dhamdhere

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

7Computer Science Department

Kedar Dhamdhere

Stochastic MST

Today Tomorrow

Prob = 1/4

Prob = 1/2

Prob = 1/4

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Kedar Dhamdhere

Stochastic MST

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

Prob = 1/4

Prob = 1/2

Prob = 1/4

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Kedar Dhamdhere

The goal• Approximation algorithm under the scenario

model

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

scenarios

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Kedar Dhamdhere

The goal• Approximation algorithm under the scenario

model

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

scenarios

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Kedar Dhamdhere

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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Kedar Dhamdhere

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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Kedar Dhamdhere

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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Kedar Dhamdhere

LP formulation

min e c0e x0

e+ i pi (e cie xi

e)

s.t.e 2 S x0

e+ xie ¸ 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: x0e, xi

e

• 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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Kedar Dhamdhere

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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Kedar Dhamdhere

Proof idea• Lemma: Cost paid in each round is at most

OPT

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Kedar Dhamdhere

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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Kedar Dhamdhere

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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Kedar Dhamdhere

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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Kedar Dhamdhere

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