Spanning Trees Introduction to Algorithms Spanning Trees CSE 680 Prof. Roger Crawfis.
On Stochastic Minimum Spanning Trees
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Transcript of On Stochastic Minimum Spanning Trees
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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 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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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