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Page 1: Approximation Algorithms for Stochastic Optimization

Approximation Algorithms for Stochastic Optimization

Chaitanya SwamyCaltech and U. Waterloo

Joint work with David Shmoys Cornell University

Page 2: Approximation Algorithms for Stochastic Optimization

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.

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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 – can augment earlier solution paying a recourse cost.

Choose initial (stage I) decisions to minimize

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

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2-stage problem 2 decision points

0.2

0.020.3 0.1

stage I

stage II scenarios

0.5

0.20.3

0.4

stage I

stage II

scenarios in stage k

k-stage problem k decision points

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2-stage problem 2 decision points

Choose stage I decisions to minimize expected total cost =

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

0.2

0.020.3 0.1

stage I

stage II scenarios

0.5

0.2 0.4

stage I

stage II

k-stage problem k decision points

0.3

scenarios in stage k

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2-Stage Stochastic Facility Location

Distribution over clients gives the set of clients to serve.

client set D

facility

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

Stage I cost = ∑(i opened) fi .

stage I facility

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2-Stage Stochastic Facility Location

Distribution over clients gives the set of clients to serve.

facility

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

Stage I cost = ∑(i opened) fi .

stage I facility

Actual scenario A = { clients to serve}, materializes.Stage II: Can open more facilities to serve clients in A; pay cost fi

A to open facility i. Assign clients in A to facilities.

Stage II cost = ∑ fiA + (cost of serving

clients in A).

i opened inscenario A

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Want to decide which facilities to open in stage I.

Goal: Minimize Total Cost = (stage I cost) + EA D [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.

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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 -approximation algorithm if,

•A runs in polynomial time.

•A(I) ≤ .OPT(I) on all instances I,

is called the approximation ratio of A.

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Overview of Previous Work• polynomial scenario model: Dye, Stougie &

Tomasgard;Ravi & Sinha; Immorlica, Karger, Minkoff & Mirrokni.

• Immorlica et al.: also consider independent activation modelproportional costs: (stage II cost) = (stage I cost),

e.g., fiA = .fi for 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.

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Boosted Sampling (GPRS04)

Proportional costs: (stage II cost) = (stage cost)Note: is same as in previous talk.

– Sample 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.

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Shmoys, S ’04 vs. Boosted sampling

– 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) = (stage cost) can depend on scenario.

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

Need only samples.

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

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Stochastic Set Cover (SSC)

Universe U = {e1, …, en }, subsets S1, S2, …, Sm 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 U to be covered is revealed – can pick additional sets paying wS

A for set S.

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

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A Linear Program for SSCFor simplicity, consider wS

A = WS for every scenario A.S : stage weight of set S

pA : probability of scenario A U.xS : indicates if set S is picked in stage I.yA,S : indicates if set S is picked in scenario A.Minimize ∑S SxS + ∑AU pA ∑S WSyA,S

subject to,

∑S:eS xS + ∑S:eS yA,S ≥ 1 for

each A U, eA

xS, yA,S ≥ 0 for each S, A.

Exponential number of variables and exponential number of constraints.

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A Rounding Theorem

Assume LP can be solved in polynomial time.

Suppose for the deterministic problem, we have an -approximation algorithm wrt.

the LP relaxation, i.e., A such that A(I) ≤ .

(optimal LP solution for 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 an -approx. algorithm to get a 2-approximation algorithm for stochastic set cover.

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Rounding the LPAssume LP can be solved in polynomial time. Suppose we have an -approximation algorithm wrt. the LP relaxation for the deterministic problem.

Let E = {e : ∑S:eS xS ≥ ½}.

So (2x) is a fractional set cover for the set E can round to get an integer set cover S for E of cost ∑SS S ≤ (∑S 2SxS) .

S is the first stage decision.

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

∑S:eS xS + ∑S:eS yA,S ≥ 1 for each A U, eA

for every element e, either

∑S:eS xS ≥ ½ OR in each scenario A : eA, ∑S:eS yA,S ≥ ½.

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Sets

Elements

Rounding (contd.)

Set in S

Element in E

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

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

So (2yA) is a fractional set cover for A\E can round to get a set cover of W-cost ≤ (∑S

2WSyA,S) .

A

Using this to augment S in scenario A, expected cost

≤ ∑SS S +

2.∑ AU pA (∑S WSyA,S) ≤ 2.OPT.

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An -approx. algorithm for deterministic problem gives a 2-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 (contd.)

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Let E = {e : ∑S:eS xS ≥ ½}.

So (2x) is a fractional set cover for the set E can round to get an integer set cover S of cost ∑SS S ≤ (∑S 2SxS) .

S is the first stage decision.

Rounding the LPAssume LP can be solved in polynomial time. Suppose we have an -approximation algorithm wrt. the LP relaxation for the deterministic problem.

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

∑S:eS xS + ∑S:eS yA,S ≥ 1 for each A U, eA

for every element e, either

∑S:eS xS ≥ ½ OR in each scenario A : eA, ∑S:eS yA,S ≥ ½.

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A Compact Convex Program

pA : probability of scenario A U.

xS : indicates if set S is picked in stage I.Minimize h(x) = ∑S SxS + ∑AU pAfA(x) s.t. xS ≥ 0

for each S (SSC-

P)

where fA(x) = min. ∑S WSyA,S

s.t. ∑S:eS yA,S ≥ 1 – ∑S:eS xS for each eA

yA,S ≥ 0 for each S.

Equivalent to earlier LP.

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

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The General Strategy

1.Get a (1+)-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 -approx. algorithm for the deterministic

problem to solve subproblems.

Obtain a c.-approximation algorithm for the stochastic integer problem.Many applications: set cover, vertex cover, facility location, multicut on trees, …

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•Need a procedure that at any point y,if yP, returns a violated inequality which shows that yP

Solving the Convex Program

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

Ellipsoid method Py

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•Need a procedure that at any point y,if yP, returns a violated inequality which shows that yPif yP, computes the subgradient of h(.) at y

dm is a subgradient of 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, …, xkP such that mini=1…k h(xi) is close to OPT.

Solving the Convex Program

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

Ellipsoid method

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

P

yh(x) ≤ h(y)

d

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•Need a procedure that at any point y,if yP, returns a violated inequality which shows that yPif yP, computes an approximate subgradient of h(.) at y

d'm is an -subgradient at u,

if vP, h(v)-h(u) ≥ d'.(v-u) – .h(u).

•Given such a procedure, can compute point xP such thath(x) ≤ OPT/(1-) + without ever evaluating h(.)!

Solving the Convex Program

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

P

yh(x) ≤ h(y)

d'

Ellipsoid method

Can compute -subgradients by sampling.

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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+).OPT guarantee.

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

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

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A Solvable Class of Stochastic LPs

Minimize h(x) = .x + ∑AU pAfA(x) s.t. x P

m

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

s.t. DA rA + TA

yA ≥ jA – TA x

yA m, rA n, yA, rA ≥ 0.Theorem: Can get a (1+)-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.

≥ 0

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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 )!

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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+)-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 -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.

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Multi-stage ProblemsGiven :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 – can augment earlier solution paying a recourse cost.

k-stage problem k decision points

0.5

0.2 0.4

stage I

stage II

scenarios in stage k Choose stage I decisions to minimize

expected total cost = (stage I cost) + Eall scenarios [cost of stages 2 …

k].

0.3

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Multi-stage ProblemsFix k = number of stages.LP-rounding: S, Shmoys ’05– Ellipsoid-based algorithm extends– SAA method also works

black-box model, arbitrary costsRounding 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 -subgradients is significantly harder, need several new ideas

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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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Thank You.


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