on stochastic minimum spanning trees kedar dhamdhere computer science department joint work with:...
Post on 21-Dec-2015
213 views
TRANSCRIPT
On Stochastic Minimum Spanning Trees
Kedar DhamdhereComputer Science Department
Joint work with: Mohit Singh, R. Ravi (IPCO 05)
2Computer Science Department
Kedar Dhamdhere
Outline
• Stochastic Optimization Model• Related Work• Algorithm for Stochastic MST• Conclusion
3Computer Science Department
Kedar 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
4Computer Science Department
Kedar 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
5Computer Science Department
Kedar 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
6Computer Science Department
Kedar 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
7Computer Science Department
Kedar Dhamdhere
Stochastic MST
Today Tomorrow
Prob = 1/4
Prob = 1/2
Prob = 1/4
8Computer Science Department
Kedar Dhamdhere
Stochastic MST
Today’s cost = 2 Tomorrow’s E[cost] = 1
Prob = 1/4
Prob = 1/2
Prob = 1/4
9Computer Science Department
Kedar Dhamdhere
The goal
• Approximation algorithm under the scenario model
• NP-hardness • Probability distribution given as a set of
scenarios
10Computer Science Department
Kedar Dhamdhere
The goal
• Approximation algorithm under the scenario model
• NP-hardness • Probability distribution given as a set of
scenarios
11Computer Science Department
Kedar 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]
12Computer Science Department
Kedar 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
13Computer Science Department
Kedar 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
14Computer Science Department
Kedar Dhamdhere
LP formulation
min e c0e x0
e+ i pi (e cie xi
e)
s.t.
e 2 S x0e+ xi
e ¸ 1 8 S ½ 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
15Computer Science Department
Kedar Dhamdhere
Algorithm: randomized rounding
• Solve the LP formulation– fractional solution: x0
e, xie
• For O(log nk) rounds– Include an edge independent of others in the first
stage solution with probability x0e
– Include an edge independent of others in the ith scenario with probability xi
e
20Computer Science Department
Kedar Dhamdhere
Proof idea
• Lemma: Cost paid in each round is at most OPT
21Computer Science Department
Kedar 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
22Computer Science Department
Kedar 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.
23Computer Science Department
Kedar Dhamdhere
Other models for second stage costs
• Sampling 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 changes– Sample poly(n,) scenarios from “black box”
24Computer Science Department
Kedar Dhamdhere
Other models for second stage costs
• Independent 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