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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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2 Computer Science DepartmentKedar Dhamdhere Outline Stochastic Optimization Model Related Work Algorithm for Stochastic MST Conclusion

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3 Computer Science DepartmentKedar Dhamdhere 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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4 Computer Science DepartmentKedar Dhamdhere 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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5 Computer Science DepartmentKedar Dhamdhere 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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6 Computer Science DepartmentKedar Dhamdhere 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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7 Computer Science DepartmentKedar Dhamdhere Stochastic MST Today Tomorrow Prob = 1/4 Prob = 1/2 Prob = 1/4

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8 Computer Science DepartmentKedar Dhamdhere Stochastic MST Todays cost = 2 Tomorrows E[cost] = 1 Prob = 1/4 Prob = 1/2 Prob = 1/4

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9 Computer Science DepartmentKedar Dhamdhere The goal Approximation algorithm under the scenario model NP-hardness Probability distribution given as a set of scenarios

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10 Computer Science DepartmentKedar Dhamdhere The goal Approximation algorithm under the scenario model NP-hardness Probability distribution given as a set of scenarios

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11 Computer Science DepartmentKedar Dhamdhere 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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12 Computer Science DepartmentKedar Dhamdhere 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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13 Computer Science DepartmentKedar Dhamdhere 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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14 Computer Science DepartmentKedar Dhamdhere LP formulation min e c 0 e x 0 e + i p i ( e c i e x i e ) s.t. e 2 S x 0 e + x i e 1 8 S V, 1 i k x i e 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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15 Computer Science DepartmentKedar Dhamdhere Algorithm: randomized rounding Solve the LP formulation fractional solution: x 0 e, x i e For O(log nk) rounds Include an edge independent of others in the first stage solution with probability x 0 e Include an edge independent of others in the i th scenario with probability x i e

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16 Computer Science DepartmentKedar Dhamdhere Example Today Tomorrow

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17 Computer Science DepartmentKedar Dhamdhere Example: round 1 Today Tomorrow

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18 Computer Science DepartmentKedar Dhamdhere Example: round 1 Today Tomorrow

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19 Computer Science DepartmentKedar Dhamdhere Example: round 2 Today Tomorrow

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20 Computer Science DepartmentKedar Dhamdhere Proof idea Lemma: Cost paid in each round is at most OPT

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21 Computer Science DepartmentKedar Dhamdhere 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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22 Computer Science DepartmentKedar Dhamdhere 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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23 Computer Science DepartmentKedar Dhamdhere Other models for second stage costs Sampling Access: Black box available which generates a sample of 2 nd 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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24 Computer Science DepartmentKedar Dhamdhere Other models for second stage costs Independent costs: second stage cost 2 u.a.r [0,1] Threshold heuristic with performance guarantee OPT + (3)/4 [Frieze 85] Single stage costs 2 u.a.r [0,1]; MST has cost (3) [Flaxman et al. 05] Both stage costs 2 u.a.r [0,1]; Thresholding heuristic gives cost (3) 1/2

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25 Computer Science DepartmentKedar Dhamdhere Conclusions Tight approximation algorithm for stochastic MST based on randomized rounding Extensions to other models for uncertainty in data