Approximation algorithms for stochastic optimization
This presentation is the property of its rightful owner.
Sponsored Links
1 / 32

Approximation Algorithms for Stochastic Optimization PowerPoint PPT Presentation


  • 61 Views
  • Uploaded on
  • Presentation posted in: General

Approximation Algorithms for Stochastic Optimization. Chaitanya Swamy Caltech and U. Waterloo Joint work with David Shmoys Cornell University. Stochastic Optimization. Way of modeling uncertainty .

Download Presentation

Approximation Algorithms for Stochastic Optimization

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Approximation algorithms for stochastic optimization

Approximation Algorithms for Stochastic Optimization

Chaitanya Swamy

Caltech and U. Waterloo

Joint work with David Shmoys Cornell University


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.


Stochastic recourse models

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).


Approximation algorithms for stochastic optimization

2-stage problem

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


Approximation algorithms for stochastic optimization

2-stage problem

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].


2 stage stochastic facility location

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 Location

Distribution over clients gives the set of clients to serve.

facility

client set D


2 stage stochastic facility location1

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 Location

Distribution 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


Approximation algorithms for stochastic optimization

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

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

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

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

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

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

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 ≥ 0for each S, A.

Exponential number of variables and exponential number of constraints.


A rounding theorem

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

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.


Rounding contd

A

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 contd1

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 lp1

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

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 ≥ 0for 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

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

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 program1

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 program2

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).

  • 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

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

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

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

  • 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

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 problems1

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

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?


Thank you

Thank You.


  • Login