On Stochastic Minimum Spanning Trees

1
On Stochastic Minimum Spanning Trees

• Kedar Dhamdhere
• Computer Science Department
• Joint work with Mohit Singh, R. Ravi (IPCO 05)

2
Outline

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

3
Stochastic optimization

• Classical optimization assumes deterministic
  inputs
• Real world data has uncertainties
• Dantzig 55, Beale 61 Modeling data
  uncertainty as probability distribution over
  inputs

4
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

5
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

6
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
Prob 1/4
Prob 1/2
Prob 1/4
Today
Tomorrow
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Stochastic MST
Prob 1/4
Prob 1/2
Prob 1/4
Todays cost 2
Tomorrows Ecost 1
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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

12
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 minlog n, log k-hard to
  approximate unless P NP

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LP formulation

• min ?e c0e x0e ?i pi (?e cie xie)
• s.t.
• ?e 2 S x0e 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, 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 xie

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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