Download  On Stochastic Minimum Spanning Trees

On Stochastic Minimum Spanning Trees
Kedar DhamdhereComputer Science Department
Joint work with: Mohit Singh, R. Ravi (IPCO 05)
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OutlineStochastic Optimization ModelRelated WorkAlgorithm for Stochastic MSTConclusion
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Stochastic optimizationClassical optimization assumes deterministic inputsReal 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] Twostage stochastic opt. with recourse
Two stages of decision makingProbability dist. governing second stage data and costsSolution can always be made feasible in second stage
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Common framework[Birge, Louveaux 97] Twostage stochastic opt. with recourse
Two stages of decision makingProbability dist. governing second stage data and costsSolution can always be made feasible in second stage
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Common framework[Birge, Louveaux 97] Twostage stochastic opt. with recourse
Two stages of decision makingProbability dist. governing second stage data and costsSolution can always be made feasible in second stage
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Stochastic MSTTodayTomorrowProb = 1/4Prob = 1/2Prob = 1/4
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Stochastic MSTTodays cost = 2Tomorrows E[cost] = 1Prob = 1/4Prob = 1/2Prob = 1/4
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The goalApproximation algorithm under the scenario model
NPhardness Probability distribution given as a set of scenarios
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The goalApproximation algorithm under the scenario model
NPhardness Probability distribution given as a set of scenarios
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Related workStochastic 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]2approximation for stochastic covering problem given approximation for the deterministic problem
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Our results: approximation algorithmTheorem: 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 formulationmin e c0e x0e+ i pi (e cie xie)s.t. e 2 S x0e+ xie 1 8 S 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 roundingSolve the LP formulationfractional solution: x0e, xie
For O(log nk) roundsInclude an edge independent of others in the first stage solution with probability x0eInclude an edge independent of others in the ith scenario with probability xie
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ExampleTodayTomorrow
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Example: round 1TodayTomorrow
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Example: round 1TodayTomorrow
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Example: round 2TodayTomorrow
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Proof ideaLemma: Cost paid in each round is at most OPT
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Proof ideaLemma: 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 9/10At least one edge leaving a component is included with prob 0.63
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Proof ideaLemma: 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 9/10At 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 costsSampling 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 changesSample poly(n,) scenarios from black box
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Other models for second stage costsIndependent 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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ConclusionsTight approximation algorithm for stochastic MST based on randomized rounding
Extensions to other models for uncertainty in data
animateChange tomorrows costsAdd probabilitiesanimateEmphasize: why approx algorithm complexityScenario model way to model prob distributionEmphasize: why approx algorithm complexityScenario model way to model prob distributionEdge in TimeMention that ShmoysSwamys covering program does not work.This LP can be solved efficiently using ellipsoidContrast this with Set Cover: log(#cuts) approximation for setcover#elements here is 2^n.Its clear that the cost paid in each round is at most OPT