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

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

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

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

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

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

7. 7 Computer Science DepartmentKedar Dhamdhere Stochastic MST Today Tomorrow Prob = 1/4 Prob = 1/2 Prob = 1/4

8. 8 Computer Science DepartmentKedar Dhamdhere Stochastic MST Today’s cost = 2 Tomorrow’s E[cost] = 1 Prob = 1/4 Prob = 1/2 Prob = 1/4

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

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

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]

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

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

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 ½ 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

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

16. 16 Computer Science DepartmentKedar Dhamdhere Example Today Tomorrow

17. 17 Computer Science DepartmentKedar Dhamdhere Example: round 1 Today Tomorrow

18. 18 Computer Science DepartmentKedar Dhamdhere Example: round 1 Today Tomorrow

19. 19 Computer Science DepartmentKedar Dhamdhere Example: round 2 Today Tomorrow

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

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

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.

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”

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

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