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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


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


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








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


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


The goalApproximation algorithm under the scenario model

NP-hardness Probability distribution given as a set of scenarios


The goalApproximation algorithm under the scenario model

NP-hardness Probability distribution given as a set of scenarios


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]


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


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


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


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


ExampleTodayTomorrow


Example: round 1TodayTomorrow






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



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.


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


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


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

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