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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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• Slide 1
• On Stochastic Minimum Spanning Trees Kedar Dhamdhere Computer Science Department Joint work with: Mohit Singh, R. Ravi (IPCO 05)
• Slide 2
• 2 Computer Science DepartmentKedar Dhamdhere Outline Stochastic Optimization Model Related Work Algorithm for Stochastic MST Conclusion
• Slide 3
• 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
• Slide 4
• 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
• Slide 5
• 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
• Slide 6
• 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
• Slide 7
• 7 Computer Science DepartmentKedar Dhamdhere Stochastic MST Today Tomorrow Prob = 1/4 Prob = 1/2 Prob = 1/4
• Slide 8
• 8 Computer Science DepartmentKedar Dhamdhere Stochastic MST Todays cost = 2 Tomorrows E[cost] = 1 Prob = 1/4 Prob = 1/2 Prob = 1/4
• Slide 9
• 9 Computer Science DepartmentKedar Dhamdhere The goal Approximation algorithm under the scenario model NP-hardness Probability distribution given as a set of scenarios
• Slide 10
• 10 Computer Science DepartmentKedar Dhamdhere The goal Approximation algorithm under the scenario model NP-hardness Probability distribution given as a set of scenarios
• Slide 11
• 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]
• Slide 12
• 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
• Slide 13
• 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
• Slide 14
• 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
• Slide 15
• 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
• Slide 16
• 16 Computer Science DepartmentKedar Dhamdhere Example Today Tomorrow
• Slide 17
• 17 Computer Science DepartmentKedar Dhamdhere Example: round 1 Today Tomorrow
• Slide 18
• 18 Computer Science DepartmentKedar Dhamdhere Example: round 1 Today Tomorrow
• Slide 19
• 19 Computer Science DepartmentKedar Dhamdhere Example: round 2 Today Tomorrow
• Slide 20
• 20 Computer Science DepartmentKedar Dhamdhere Proof idea Lemma: Cost paid in each round is at most OPT
• Slide 21
• 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
• Slide 22
• 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.
• Slide 23
• 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
• Slide 24
• 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
• Slide 25
• 25 Computer Science DepartmentKedar Dhamdhere Conclusions Tight approximation algorithm for stochastic MST based on randomized rounding Extensions to other models for uncertainty in data