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Approximation Algorithms for Stochastic Optimization

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### Approximation Algorithms for Stochastic Optimization

Chaitanya Swamy

Caltech and U. Waterloo

Joint work with David Shmoys Cornell University

Stochastic Optimization

- Way of modeling uncertainty.
- Exact data is unavailable or expensive – data is uncertain, specified by a probability distribution.
Want to make the best decisions given this uncertainty in the data.

- Applications in logistics, transportation models, financial instruments, network design, production planning, …
- Dates back to 1950’s and the work of Dantzig.

Stochastic Recourse Models

Given : Probability distribution over inputs.

Stage I : Make some advance decisions – plan ahead or hedge against uncertainty.

Uncertainty evolves through various stages.

Learn new information in each stage.

Can take recourse actions in each stage – canaugment earlier solution paying a recourse cost.

Choose initial (stage I) decisions to minimize

(stage I cost) + (expected recourse cost).

º 2 decision points

k-stage problem

º k decision points

stage I

0.2

0.4

0.3

stage II

0.5

scenarios in stage k

stage I

0.2

0.02

0.3

0.1

stage IIscenarios

º 2 decision points

k-stage problem

º k decision points

stage I

stage I

0.2

0.2

0.4

0.3

stage II

0.02

0.3

0.1

0.5

stage IIscenarios

scenarios in stage k

Choose stage I decisions to minimize

expected total cost =

(stage I cost) + Eall scenarios [cost of stages 2 … k].

Stage I: Open some facilities in advance; pay cost fifor facility i.

Stage I cost = ∑(i opened) fi.

stage I facility

2-Stage Stochastic Facility LocationDistribution over clients gives the set of clients to serve.

facility

client set D

Actual scenarioA = { clients to serve}, materializes.

Stage II: Can open more facilities to serve clients in A; pay cost fiA to open facility i. Assign clients in A to facilities.

Stage II cost = ∑ fiA + (cost of serving clients in A).

i opened in

scenario A

2-Stage Stochastic Facility LocationDistribution over clients gives the set of clients to serve.

Stage I: Open some facilities in advance; pay cost fifor facility i.

Stage I cost = ∑(i opened) fi.

facility

stage I facility

Want to decide which facilities to open in stage I.

Goal: Minimize Total Cost =

(stage I cost)+ EA ÍD[stage II cost for A].

- How is the probability distribution specified?
- A short (polynomial) list of possible scenarios
- Independent probabilities that each client exists
- A black box that can be sampled.

Approximation Algorithm

- Hard to solve the problem exactly.
- Even special cases are #P-hard.
- Settle for approximate solutions. Give polytime algorithm that always finds near-optimal solutions.
- A is a a-approximation algorithm if,
- A runs in polynomial time.
- A(I) ≤ a.OPT(I) on all instances I,
- ais called the approximation ratio of A.

Overview of Previous Work

- polynomial scenario model: Dye, Stougie & Tomasgard;
- Ravi & Sinha; Immorlica, Karger, Minkoff & Mirrokni.
- Immorlica et al.: also consider independent activation model
proportional costs: (stage II cost) = l(stage I cost),

- e.g., fiA = l.fifor each facility i,in each scenario A.
- Gupta, Pál, Ravi & Sinha (GPRS04): black-box model but also with proportional costs.
- Shmoys, S (SS04): black-box model with arbitrary costs.
- approximation scheme for 2-stage LPs + rounding procedure “reduces” stochastic problems to their deterministic versions.
- for some problems improve upon previous results.

Boosted Sampling (GPRS04)

- Proportional costs: (stage II cost) = l(stage I cost)
- Note: l is same as s in previous talk.
- Sample l times from distribution
- Use “suitable” algorithm to solve deterministic instance consisting of sampled scenarios (e.g., all sampled clients) – determines stage I decisions

- Analysis relies on the existence of cost-shares that can be used to share the stage I cost among sampled scenarios.

Shmoys, S ’04 vs. Boosted sampling

Both work in the black-box model: arbitrary distributions.

- Can handle arbitrary costs in the two stages.
- LP rounding: give an algorithm to solve the stochastic LP.
- Need many more samples to solve stochastic LP.

Need proportional costs:

(stage II cost) = l(stage I cost)

l can depend on scenario.

Primal-dual approach: cost-shares obtained by exploiting structure via primal-dual schema.

Need only l samples.

Stochastic Set Cover (SSC)

Universe U = {e1, …, en }, subsets S1, S2, …, SmÍ U, set S has weight wS.

Deterministic problem: Pick a minimum weight collection of sets that covers each element.

- Stochastic version: Set of elements to be covered is given by a probability distribution.
- choose some sets initially paying wS for set S
- subset A Í U to be covered is revealed
- can pick additional sets paying wSAfor set S.

- Minimize(w-cost of sets picked in stage I)+
- EA ÍU [wA-cost of new sets picked for scenario A].

A Linear Program for SSC

For simplicity, consider wSA = WS for every scenario A.

wS : stage I weight of set S

pA : probability of scenario A Í U.

xS : indicates if set S is picked in stage I.

yA,S: indicates if set S is picked in scenario A.

Minimize ∑SwSxS +∑AÍU pA ∑S WSyA,S

subject to,

∑S:eÎS xS + ∑S:eÎS yA,S ≥ 1for each A Í U, eÎA

xS, yA,S ≥ 0 for each S, A.

Exponential number of variables and exponential number of constraints.

A Rounding Theorem

Assume LP can be solved in polynomial time.

Suppose for the deterministic problem, we have an a-approximation algorithm wrt. the LP relaxation, i.e., A such that A(I)≤a.(optimal LP solutionfor I)

for every instance I.

e.g., “the greedy algorithm” for set cover is a

log n-approximation algorithm wrt. LP relaxation.

Theorem: Can use such ana-approx. algorithm to get a 2a-approximation algorithm for stochastic set cover.

Rounding the LP

Assume LP can be solved in polynomial time.

Suppose we have an a-approximation algorithm wrt. the LP relaxation for the deterministic problem.

Let (x,y) : optimal solution with cost OPT.

∑S:eÎS xS + ∑S:eÎS yA,S ≥ 1for each A Í U, eÎA

Þ for every element e, either

∑S:eÎS xS ≥ ½ OR in each scenario A : eÎA, ∑S:eÎS yA,S ≥ ½.

Let E = {e : ∑S:eÎS xS ≥ ½}.

So (2x) is afractional set coverfor the set E Þ can round to get aninteger set coverSfor E of cost ∑SÎSwS ≤ a(∑S 2wSxS) .

Sis the first stage decision.

Consider any scenario A. Elements in A Ç E are covered.

For every e Î A\E, it must be that ∑S:eÎS yA,S ≥ ½.

So (2yA) is afractional set coverfor A\E Þ can round to get a set coverof W-cost≤a(∑S 2WSyA,S) .

Rounding (contd.)Sets

Set in S

Elements

Element in E

Using thisto augmentSin scenario A,expected cost

≤ ∑SÎSwS+ 2a.∑ AÍU pA (∑S WSyA,S) ≤ 2a.OPT.

Rounding (contd.)

- An a-approx. algorithm for deterministic problem gives a 2a-approximation guarantee for stochastic problem.
- In the polynomial-scenario model, gives simple polytime approximation algorithms for covering problems.
- 2log n-approximation for SSC.
- 4-approximation for stochastic vertex cover.
- 4-approximation for stochastic multicut on trees.

- Ravi & Sinha gave a log n-approximation algorithm for SSC, 2-approximation algorithm for stochastic vertex cover in the polynomial-scenario model.

Rounding the LP

Assume LP can be solved in polynomial time.

Suppose we have an a-approximation algorithm wrt. the LP relaxation for the deterministic problem.

Let (x,y) : optimal solution with cost OPT.

∑S:eÎS xS + ∑S:eÎS yA,S ≥ 1for each A Í U, eÎA

Þ for every element e, either

∑S:eÎS xS ≥ ½ OR in each scenario A : eÎA, ∑S:eÎS yA,S ≥ ½.

Let E = {e : ∑S:eÎS xS ≥ ½}.

So (2x) is afractional set coverfor the set E Þ can round to get aninteger set coverSof cost ∑SÎSwS ≤ a(∑S 2wSxS) .

Sis the first stage decision.

A Compact Convex Program

pA : probability of scenario A Í U.

xS : indicates if set S is picked in stage I.

Minimize h(x) = ∑SwSxS +∑AÍU pAfA(x) s.t. xS ≥ 0 for each S

(SSC-P)

where fA(x) = min. ∑S WSyA,S

s.t. ∑S:eÎS yA,S ≥ 1– ∑S:eÎS xSfor each eÎA

yA,S ≥ 0 for each S.

Equivalent to earlier LP.

Each fA(x) is convex, so h(x) is a convex function.

The General Strategy

1. Get a (1+e)-optimal fractional first-stage solution (x) by solving the convex program.

2. Convert fractional solution (x) to integer solution

- decouple the two stages near-optimally
- use a-approx. algorithm for the deterministic problem to solve subproblems.

Obtain a c.a-approximation algorithm for the stochastic integer problem.

Many applications: set cover, vertex cover, facility location, multicut on trees, …

Solving the Convex Program

Minimize h(x) subject to xÎP. h(.) : convex

Ellipsoid method

P

y

- Need a procedure that at any point y,
- if yÏP, returns a violated inequality
- which shows that yÏP

Solving the Convex Program

Minimize h(x) subject to xÎP. h(.) : convex

Ellipsoid method

P

- Need a procedure that at any point y,
- if yÏP, returns a violated inequality
- which shows that yÏP
- if yÎP, computes the subgradient
- of h(.) at y
- dÎÂm is a subgradientof h(.) at u, if "v, h(v)-h(u) ≥ d.(v-u).

- Given such a procedure, ellipsoid runs in polytime and returns points x1, x2, …, xkÎP such that mini=1…k h(xi) is close to OPT.

h(x) ≤ h(y)

y

d

Computing subgradients is hard. Evaluating h(.) is hard.

Solving the Convex Program

Minimize h(x) subject to xÎP. h(.) : convex

Ellipsoid method

P

- Need a procedure that at any point y,
- if yÏP, returns a violated inequality
- which shows that yÏP
- if yÎP, computes an approximate
- subgradient of h(.) at y
- d'ÎÂm is an e-subgradient at u,
- if "vÎP, h(v)-h(u) ≥ d'.(v-u) – e.h(u).

- d'ÎÂm is an e-subgradient at u,

- Given such a procedure, can compute point xÎP such that
- h(x) ≤OPT/(1-e) + r without ever evaluating h(.)!

h(x) ≤ h(y)

y

d'

Can compute e-subgradients by sampling.

Putting it all together

Get solution x with h(x) close to OPT.

Sample initially to detect if OPT is large – this allows one to get a (1+e).OPT guarantee.

Theorem: (SSC-P) can be solved to within a factor of(1+e)in polynomial time, with high probability.

Gives a(2log n+e)-approximation algorithm forthe stochastic set cover problem.

A Solvable Class of Stochastic LPs

Minimize h(x) = w.x+∑AÍU pAfA(x) s.t. x ÎPÍÂm

where fA(x) = min. wA.yA + cA.rA

s.t. DArA + TAyA ≥ jA – TAx

yA Î Âm, rA Î Ân, yA, rA ≥ 0.

≥ 0

Theorem: Can get a (1+e)-optimal solution for this class of stochastic programs in polynomial time.

Includes covering problems (e.g., set cover, network design, multicut), facility location problems, multicommodity flow.

Moral of the Story

- Even though the stochastic LP relaxation has exponentially many variables and constraints, we can still obtain near-optimal fractional first-stage decisions
- Fractional first-stage decisions are sufficient to decouple the two stages near-optimally
- Many applications: set cover, vertex cover, facility locn., multicommodity flow, multicut on trees, …
- But we have to solve convex program with many samples (not just l)!

Sample Average Approximation

- Sample Average Approximation (SAA) method:
- Sample initially N times from scenario distribution
- Solve 2-stage problem estimating pA with frequency of occurrence of scenario A

- How large should N be to ensure that an optimal solution to sampled problem is a (1+e)-optimal solution to original problem?

- Kleywegt, Shapiro & Homem De-Mello (KSH01):
- bound N by variance of a certain quantity – need not be polynomially bounded even for our class of programs.

- S, Shmoys ’05 :
- show using e-subgradients that for our class, N can be poly-bounded.

- Charikar, Chekuri & Pál ’05:
- give another proof that for a class of 2-stage problems, N can be poly-bounded.

Multi-stage Problems

k-stage problem

º k decision points

Given : Distribution over inputs.

Stage I : Make some advance decisions – hedge against uncertainty.

Uncertainty evolves in various stages.

Learn new information in each stage.

Can take recourse actions in each stage – canaugment earlier solution paying a recourse cost.

stage I

0.2

0.4

0.3

stage II

0.5

scenarios in stage k

Choose stage I decisions to minimize

expected total cost =

(stage I cost) + Eall scenarios [cost of stages 2 … k].

Multi-stage Problems

Fix k = number of stages.

LP-rounding: S, Shmoys ’05

- Ellipsoid-based algorithm extends
- SAA method also works
black-box model, arbitrary costs

Rounding procedure of SS04 can be easily adapted: lose an O(k)-factor over the deterministic guarantee

- O(k)-approx. for k-stage vertex cover, facility location, multicut on trees; k.log n-approx. for k-stage set cover
Gupta, Pál, Ravi & Sinha ’05: boosted sampling extends but with outcome-dependent proportional costs

- 2k-approx. for k-stage Steiner tree (also Hayrapetyan, S & Tardos)
- factors exponential in k for k-stage vertex cover, facility location

Computing e-subgradients is significantly harder, need several new ideas

Open Questions

- Combinatorial algorithms in the black box model and with general costs. What about strongly polynomial algorithms?
- Incorporating “risk” into stochastic models.
- Obtaining approximation factors independent of k for k-stage problems.
Integrality gap for covering problems does not increase. Munagala has obtained a 2-approx. for k-stage VC.

- Is there a larger class of doubly exponential LPs that one can solve with (more general) techniques?

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