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On Stochastic Minimum Spanning Trees Kedar Dhamdhere Computer Science Department Joint work with: Mohit Singh, R. Ravi (IPCO 05)

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Kedar Dhamdhere Computer Science Department Joint work with: Mohit Singh, R. Ravi (IPCO 05). On Stochastic Minimum Spanning Trees. Outline. Stochastic Optimization Model Related Work Algorithm for Stochastic MST Conclusion. Stochastic optimization. - PowerPoint PPT Presentation

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  • On Stochastic Minimum Spanning Trees

    Kedar DhamdhereComputer Science Department

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

    *Computer Science Department

    OutlineStochastic Optimization ModelRelated WorkAlgorithm for Stochastic MSTConclusion

    *Computer Science Department

    Stochastic optimizationClassical optimization assumes deterministic inputsReal world data has uncertainties[Dantzig 55, Beale 61] Modeling data uncertainty as probability distribution over inputs

    *Computer Science Department

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

    Two stages of decision makingProbability dist. governing second stage data and costsSolution can always be made feasible in second stage

    *Computer Science Department

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

    Two stages of decision makingProbability dist. governing second stage data and costsSolution can always be made feasible in second stage

    *Computer Science Department

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

    Two stages of decision makingProbability dist. governing second stage data and costsSolution can always be made feasible in second stage

    *Computer Science Department

    Stochastic MSTTodayTomorrowProb = 1/4Prob = 1/2Prob = 1/4

    *Computer Science Department

    Stochastic MSTTodays cost = 2Tomorrows E[cost] = 1Prob = 1/4Prob = 1/2Prob = 1/4

    *Computer Science Department

    The goalApproximation algorithm under the scenario model

    NP-hardness Probability distribution given as a set of scenarios

    *Computer Science Department

    The goalApproximation algorithm under the scenario model

    NP-hardness Probability distribution given as a set of scenarios

    *Computer Science Department

    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]

    *Computer Science Department

    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

    *Computer Science Department

    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

    *Computer Science Department

    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

    *Computer Science Department

    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

    *Computer Science Department

    ExampleTodayTomorrow

    *Computer Science Department

    Example: round 1TodayTomorrow

    *Computer Science Department

    Example: round 1TodayTomorrow

    *Computer Science Department

    Example: round 2TodayTomorrow

    *Computer Science Department

    Proof ideaLemma: Cost paid in each round is at most OPT

    *Computer Science Department

    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

    *Computer Science Department

    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.

    *Computer Science Department

    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

    *Computer Science Department

    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

    *Computer Science Department

    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 Shmoys-Swamys covering program does not work.This LP can be solved efficiently using ellipsoidContrast this with Set Cover: log(#cuts) approximation for set-cover#elements here is 2^n.Its clear that the cost paid in each round is at most OPT