24 Feb 04 Boosted Sampling 1
Boosted Sampling: Approximation Algorithms for Stochastic
Problems
Martin Pál
Joint work withAnupam Gupta R. Ravi
Amitabh Sinha
24 Feb 04 Boosted Sampling 2
Infrastructure Design Problems
Build a solution Sol of minimal cost, so that every user is satisfied.
minimize cost(Sol)
subject to satisfied(j,Sol) for j=1, 2, …, n
For example, Steiner tree:
Sol: set of edges to build
satisfied(j,Sol) iff there is a path from terminal j to root
cost(Sol) = eSol ce
24 Feb 04 Boosted Sampling 3
Infrastructure Design Problems
Assumption: Sol is a set of elements
cost(Sol) = elemSol cost(elem)
Facility location: satisfied(j) iff j connected to an open facility
Vertex Cover: satisfied(e={uv}) iff u or v in the cover
Steiner network: satisfied(j) iff j’s terminals connected by a path
Cut problems: satisfied(j) iff j’s terminals disconnected
24 Feb 04 Boosted Sampling 4
Dealing with uncertainity
Often, we do not know the exact requirements of users.
Building in advance reduces cost – but we do not have enough information.
As time progresses, we gain more information about the demands – but building under time pressure is costly.
Tradeoff between information and cost.
24 Feb 04 Boosted Sampling 5
The model
Two stage stochastic model with recourse:
On Monday, elements are cheap, but we do not know how many/which clients will show up. We can buy some elements.
On Tuesday, clients show up. Elements are now more expensive (by an inflation factor σ). We have to buy more elements to satisfy all clients.
drawn from a known distribution π
24 Feb 04 Boosted Sampling 6
The model
Two stage stochastic model with recourse:
Find Sol1 Elems and Sol2 : 2Users 2Elems to
minimize cost(Sol1) + σ Eπ(T)[cost(Sol2(T))]
subject to satisfied(j, Sol1 Sol2(T))
for all sets TUsers and all jT
Want compact representation of Sol2 by an algorithm
24 Feb 04 Boosted Sampling 7
Related work
•Stochastic linear programming dates back to works of Dantzig, Beale in the mid-50’s
•Only moderate progress on stochastic IP/MIP
•Scheduling literature, various distributions of job lengths
•Single stage stochastic: maybecast [Karger&Minkoff00], bursty connections [Kleinberg,Rabani&Tardos00]…
•Stochastic versions of NP-hard problems (restricted π) [Ravi & Sinha 03], [Immorlica, Karger, Minkoff & Mirrokni 04]
•Extensive literature on each deterministic problem
24 Feb 04 Boosted Sampling 8
Our work
•We propose a simple but powerful framework to find approximate solutions to two stage stochastic problems using approximation algorithms for their deterministic counterparts.
•For a number of problems, including Steiner Tree, Facility Location, Single Sink Rent or Buy and Steiner Forest (weaker model) our framework gives constant approximation.
•Analysis is based on strict cost sharing, developed by [Gupta,Kumar,P.&Roughgarden03]
24 Feb 04 Boosted Sampling 9
No restriction on distributions
Previous works often assume special distributions:
•Scenario model: There are k sets of users – scenarios; each scenario Ti has probability pi. [Ravi & Sinha 03].
•Independent decisions model: each client j appears with prob. pj independently of others [Immorlica et al 04].
In contrast, our scheme works for arbitrary distributions (although the independent coinflips model sometimes allows us to prove improved guarantees).
24 Feb 04 Boosted Sampling 10
The Framework
1. Boosted Sampling: Draw σ samples of clients S1,S2 ,…,Sσ from the distribution π.
2. Build the first stage solution Sol1: use Alg to build a solution for clients S = S1S2
… Sσ.3. Actual set T of clients appears. To build second stage solution Sol2, use Alg to augment Sol1 to a feasible solution for T.
Given an approx. algorithm Alg for a deterministic problem:
Example: Steiner Tree
24 Feb 04 Boosted Sampling 11
Performance Guarantee
Theorem:Let P be a sub-additive problem, with α-approximation algorithm, that admits β-strict cost sharing.
Stochastic(P) has (α+β) approx.
Corollary: Stochastic Steiner Tree, Facility Location, Vertex Cover, Steiner Network (restricted model)… have constant factor approximation algorithms.
Corollary: Deterministic and stochastic Rent-or-Buy versions of these problems have constant approximations.
24 Feb 04 Boosted Sampling 12
First Stage Cost
Recall: We - sample S1,S2 ,…,Sσ from π.
- use Alg to build solution Sol1 feasible for S=i Si
Lemma: E[cost(Sol1)] α Z*.
Pf: Let Sol = Opt1 [ Opt2(S1) … Opt2(Sσ) ].
E[cost(Sol)] cost(Opt1) + i Eπ[cost(Opt2(Si))]
= Z*.
E[cost(Sol1)] α E[cost(Sol)] (α-approximation).
Opt cost Z* = cost(Opt1) + σ Eπ[cost(Opt2(T))].
24 Feb 04 Boosted Sampling 13
Second stage cost
After Stage 2, have a solution for S’ = S1 … Sσ T.
Let Sol’ = Opt1 [ Opt2(S1) … Opt2(Sσ) Opt2(T)].
E[cost(Sol’)] cost(Opt1) + (σ+1) Eπ[cost(Opt2(Si))]
(σ+1)/σ Z*.
T is “responsible” for 1/(σ+1) part of Sol’.
If built in Stage 1, it would cost Z*/σ.
Need to build it in Stage 2 pay Z*.
Problem: do not T know when building a solution for S1 … Sσ.
24 Feb 04 Boosted Sampling 14
Idea: cost sharing
Scenario 1:
Pretend to build a solution for S’ = S T.
Charge each jS’ some amount ξ(S’,j).
Scenario 2:
Build a solution Alg(S) for S.
Augment Alg(S) to a valid solution for S’ = S T.
Assume: jS’ ξ(S’,j) Opt(S’)
We argued: E[jT ξ(S’,j)] Z*/σ (by symmetry)
Want to prove:
Augmenting cost in Scenario 2 β jT ξ(S’,j)
24 Feb 04 Boosted Sampling 15
Cost sharing function
Input: Instance of P and set of users S’
Output: cost share ξ(S,j) of each user jS’
Example: Build a spanning tree on S’ root.
Let ξ(S’,j) = cost of parental edge/2.
Note: - 2 jS’ ξ(S’,j) = cost of MST(S’)
- jS’ ξ(S’,j) cost of Steiner(S’)
24 Feb 04 Boosted Sampling 16
What properties of ξ(,) do we need?
(P1) Good approximation: cost(Alg(S)) Opt(S)
(P2) Cost shares do not overpay: jS ξ(S,j) cost(S)
(P3) Strictness: For any S,TUsers:
cost of Augment(Alg(S), T) β jT ξ(S T, j)
Second stage cost = σ cost(Augment(Alg(i Si), T)) σ β jT ξ(j Sj T, j)
E[jT ξ(j Sj T, j)] Z*/σ
Hence, E[second stage cost] σ β Z*/σ = β Z*.
24 Feb 04 Boosted Sampling 17
Strictness for Steiner Tree
Alg(S) = Min-cost spanning tree MST(S)
ξ(S,j) = cost of parental edge/2 in MST(S)
Augment(Alg(S), T):
for all jT build its parental edge in MST(S T)Alg is a 2-approx for Steiner Tree
ξ is a 2-strict cost sharing function for Alg.
Theorem: We have a 4-approx forStochastic Steiner Tree.
24 Feb 04 Boosted Sampling 18
Vertex Cover8
3
3
10 9
4
5
Users: edges
Solution: Set of vertices that covers all edges
Edge {uv} covered if at least one of u,v picked.
1
1
1 1
1
1 1
1
11
1
1 1
1
1 1
1
11
1
2 2
1
2 2
2
21
1
2 3
1
4 2
2
31
1
2 3
1
3 2
2
3
Alg: Edges uniformly raise contributions
Vertex can be paid for by neighboring edges freeze all edges adjacent to it. Buy the vertex.
Edges may be paying for both endpoints 2-approximation
Natural cost shares: ξ(S, e) = contribution of e
24 Feb 04 Boosted Sampling 19
Strictness for Vertex Cover1
1
1
1
1
n+1n+1n
S = blue edges1
1
1
1
1
T = red edge
Alg(S) = blue vertices:
Augment(Alg(S), T) costs (n+1)
ξ(S T, T) =1
•Find a better ξ? Do not know how. Instead, make Alg(S) buy a center vertex.
gap Ω(n)!
24 Feb 04 Boosted Sampling 20
Making Alg strictAlg’: - Run Alg on the same input.
- Buy all vertices that are at least 50% paid for.
1
1
1
1
1
n+1n+1n
1
1
1
1
1
½ of each vertex paid for, each edge paying for two vertices still a 4-approximation.
Augmentation (at least in our example) is free.
24 Feb 04 Boosted Sampling 21
Why should strictness hold?Alg’: - Run Alg on the same input.
- Buy all vertices that are at least 50% paid for.
Suppose vertex v fully paid for in Alg(S T).
•If jT αj’ ≥ ½ cost(v) , then T can pay for ¼ of v in the augmentation step.
•If jS αj ≥ ½ cost(v), then v would be open in Alg(S).
(almost.. need to worry that Alg(S T) and Alg(S) behave differently.)
α1
α2
α3
α1’
α2’Alg(S T) S = blue edges
T = red edgesv
24 Feb 04 Boosted Sampling 22
Metric facility location
Input: a set of facilities and a set of cities living in a metric space.
Solution: Set of open facilities, a path from each city to an open facility.“Off the shelf” components:
3-approx. algorithm [Mettu&Plaxton00].
Turns out that cost sharing fn [P.&Tardos03] is 5.45 strict.
Theorem: There is a 8.45-approx for stochastic FL.
24 Feb 04 Boosted Sampling 23
Steiner Network
client j = pair of terminals sj, tj
satisfied(j): sj, tj connected by a path2-approximation algorithms known ([Agarwal,Klein&Ravi91], [Goemans&Williamson95]), but do not admit strict cost sharing.
[Gupta,Kumar,P.,Roughgarden03]: 4-approx algorithm that admits 4-uni-strict cost sharing Theorem: 8-approx for Stochastic Steiner Network in the “independent coinflips” model.
24 Feb 04 Boosted Sampling 24
The Buy at Bulk problem
client j = pair of terminals sj, tj
Solution: an sj, tj path for j=1,…,n
cost(e) = ce f(# paths using e)
cost
# paths using e
f(e):
# paths using e
cost Rent or Buy: two pipes
Rent: $1 per path
Buy: $M, unlimited # of paths
24 Feb 04 Boosted Sampling 25
Special distributions: Rent or BuyStochastic Steiner Network:
client j = pair of terminals sj, tj
satisfied(j): sj, tj connected by a path
cost(e) = ce min(1, σ/n #paths using e)# paths using e
costn/σ
Suppose.. π({j}) = 1/n
π(S) = 0 if |S|1
Sol2({j}) is just a path!
24 Feb 04 Boosted Sampling 26
Rent or Buy
The trick works for any problem P. (can solve Rent-or-Buy Vertex Cover,..)
These techniques give the best approximation for Single-Sink Rent-or-Buy (3.55 approx [Gupta,Kumar,Roughgarden03]), and Multicommodity Rent or Buy (8-approx [Gupta,Kumar,P.,Roughgarden03], 6.83-approx [Becchetti, Konemann, Leonardi,P.04]).
“Bootstrap” to stochastic Rent-or-Buy: - 6 approximation for Stochastic Single-Sink RoB - 12 approx for Stochastic Multicommodity RoB (indep. coinflips)
24 Feb 04 Boosted Sampling 27
What if σ is also stochastic?
Suppose σ is also a random variable.
π(S, σ) – joint distribution
For i=1, 2, …, σmax do sample (Si, σi) from π with prob. σi/σmax accept Si
Let S be the union of accepted Si’s
Output Alg(S) as the first stage solution
24 Feb 04 Boosted Sampling 28
Multistage problems
Three stage stochastic Steiner Tree:
•On Monday, edges cost 1. We only know the probability distribution π.
•On Tuesday, results of a market survey come in. We gain some information I, and update π to the conditional distribution π|I. Edges cost σ1.
•On Wednesday, clients finally show up. Edges now cost σ2 (σ2>σ1), and we must buy enough to connect all clients.
Theorem: There is a 6-approximation for three stage stochastic Steiner Tree (in general, 2k approximation for k stage problem)
24 Feb 04 Boosted Sampling 29
Conclusions
We have seen a randomized algorithm for a stochastic problem: using sampling to solve problems involving randomness.
•Do we need strict cost sharing? Our proof requires strictness – maybe there is a weaker property? Maybe we can prove guarantees for arbitrary subadditive problems?
•Prove strictness for Steiner Forest – so far we have only uni-strictness.
•Cut problems: Can we say anything about Multicut? Single-source multicut?
24 Feb 04 Boosted Sampling 30
+++THE++END+++
Note that if π consists of a small number of scenarios, this can be transformed to a deterministic problem.
Find Sol1 Elems and Sol2 : 2Users 2Elems to
minimize cost(Sol1) + σ Eπ(T)[cost(Sol2(T))]
subject to satisfied(j, Sol1 Sol2(T))
for all sets TUsers and all jT
.