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Sublinear FPTASs for Stochastic Optimization Problems

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Nir Halman, HUJI

Based on joint works with D. Klabjan, C-L Lee, M. Mostagir, J. Orlin and D. Simchi-Levi

Def: An FPTAS(FullyPoly. Time Approximation Scheme) is a (1+ ε)-apx. alg. that runs in time poly. in |x| and1/ε for every instance x and ε> 0

Major techniques for FPTAS: work with dual DP + •rounding/scaling the data

•dominance - omitting states/actions dominated or approximately dominated by other states/actions

Woeginger’s framework [Wo00] uses these techniques but does not handle stochastic DP, nor exponential action spaces

- The knapsack problem
- Motivation for stochastic and oracle settings
- Approximating functions in logarithmic space and time
- Applications in the design and analysis of approximation algorithms

0/1 knapsack: Given object set {a1,…,aT} with volumes vi, profits pi, and knapsack volume B, find a subset whose total volume ≤B and total profit is maximized

DP formulation:zt(It)=max{zt+1(Ii), pt + zt+1(It –vt)}= total profit when considering items t,t+1,…,T, starting with knapsack size It.

z1(B)=?

Can be calculated in O(TB) time, i.e., pseudopolinomial in input size

NP-C, admits an FPTAS [IK75] (by scaling a dual DP)

Nonlinear knapsack: Any integer number xi < M of copies of ai can be assigned with profit pi(xi) and volume vi(xi), where pi and vi are non-decreasing functions.

Admits FPTAS when pi concave and vi convex[Ho95], and when pi is explicitly-given piecewise linear [KN09].

OPEN when pi, vi are general non-decreasing oracle functions

Fact: Some problems stop being polynomially-solvable in oracle setting (input size logM)

Def.: Given object set {a1,…,aT} with volumes vi+Di, profits pi, and knapsack volume B, find a subset whose total volume ≤B and total expected profit is maximized

1) Order of items considered is now important…

2) Benefit of adaptivity [DGV04]…

3) Fact: Some polynomially-solvable problems stop being such in stochastic setting

Def.: Given object sequence (a1,…,aT) with volumes vi+Di, profits pi, and knapsack volume B, find a subset whose total volume ≤B and total profit is maximized. Not known to admit an FPTAS

DP form.: zi(Ii) = maxx=0,…, B/viEDi{xpi+zi+1(Ii –x(vi+Di))}, (total profit to gain with remaining Ii volume by ai,…,aT) z1(B)=?

State space It=0,…,B;Action space At= 0,…, B/vt ,Profit function gt(I, x, Dt) = xpt , Transition function ft(I, x, Dt) = I– x(vt + Dt).

Observation: gt(I, x, Dt) non-decreasing in I and x;ft (I, x, Dt) non-decreasing in I, non-increasing in x. Note that At may be exponential in the input size

- The knapsack problem
- Motivation for stochastic and oracle settings
- Approximating functions in logarithmic space and time
- Applications in the design and analysis of approximation algorithms

Under what conditions an oracle function φ:D→R+ admits an efficient succinctK-approximation?(assume finite DR and K≥1. Input size = log |D|+log φmax)

Def: •φ*is aK-apx. of φif φ(x) ≤ φ*(x) ≤ Kφ(x), x (note that if φhas special structure, φ*does not necessarily)

•φ*is succinct if stored in logarithmic space and is efficient if built in logarithmic time and # of oracle queries

• φ:D→R isunimodal if x* so φ is decreasing until x* and increasing afterwards

K

K

Definition [H+08]: Let φ:D→Z+ be unimodal. A K-apx. set of φ is a subset W with argminφ, Dmax, DminW Dand the ratio between the values of φ on each two consecutive points in W is at most K

Construction: φ* is the apx. ofφinduced byW if:

φ

W

φ*

Question: How small a K-apx. ofφ can be?

Answer: Logarithmic in the input size.Moreover, it can be constructed in logarithmic time and # of oracle queries

Theorem [H+08]: If φ is either monotone, convex, or unimodal with given argmin, then it admits a succinct and efficient K-approximation function that preservesthe structure of φ

Calculus of K-apx functions [H+08]: α,β≥ 0, φi*aKi-apx. of φi(summation of apx.) α+βφ1*+φ2* is max{K1,K2}-apx. of α+βφ1+φ2 (minimization of apx.) min{φ1*,φ2*} is max{K1,K2}-apx. of min{φ1,φ2} (composition of apx.) φ1*(ψ1) is a K1-apx of φ1(ψ1) (apx. of apx.) if φ2= φ1* then φ2* is a K1K2-apx. of φ1

Optimality equation: zt(I) =min xAt(I){gt (I, x) +zt+1(ft(I, x))}

Corollary [H+08]:(mimization of summation of composition) Let gt*, z*t+1 be L1,L2-apx. functions of (unimodal) gt, zt+1 thenzt*(I)=minxAt(I){gt*(I, x)+z*t+1(ft(I, x))}is a max{L1,L2}-apx of zt(I)

Theorem [H+08]:Let Wx,1,Wx,2beKi-apx. sets of gt*, z*t+1. Then zt*(x):=minxWx,1Wx,2{g*t (It, x) +z*t+1(ft(It, x))}is a controlled (general) apx. of zt(It)

Theorem [HOS11]:φ1*aK1-apx. of φ1,φ2*aK2-apx. of φ2. If φ2cφ1for c<1/K1K2, then φ1*-φ2* is a controlled apx. of φ1-φ2

Theorem [HOS11]: Let φi*be anLi-apx. of φ1 and Wi,be aKi-apx. set of φi*, i=1,2. If φ2cφ1for c<1/K1L1L2then z*(I) =max xW1W2{φ1*(ψ1(I, x)) -φ2*(ψ2(I, x))} is a controlled apx. of z(I) =max x{φ1(ψ1(I, x)) -φ2(ψ2(I, x))}

Let F() be a CDF of integral non-negative r.v. D bounded by M, i.e., F(x)=Prob(Dx). Let

where ψis a monotone non-decreasing step function with break points a1,...,an.

We decompose ψas the sum of the 2-step functions ψ1,...,ψn, where ψi=0 for x< ai and is the constant ψ(ai)-ψ(ai-1) otherwise.

so it is the sum ofnnon-decreasing functions (n is poly., M is not)

Focuses on the domain of the functions

Unimodal functions: α,β≥ 0,WiisKi-apx. set of φi, ψ:DD,(monotonicity of apx.) every superset of W1 is a K1-apx. set of φ1 (composition of apx.) ψ -1(W1) is a K1-apx set of φ1(ψ) (linearity of apx.) W1 is a K1-apx. set of α+βφ1(maximization of apx.)W1W2 is a max{K1,K2}-apx. set of max{φ1,φ2}

Monotone functions (of the same kind):Wiis Ki-apx. set of φi, (summation of apx.) W1 W2 is a max{K1, K2}-apx. set of φ1+φ2(minimization of apx.)W1W2 is a max{K1,K2}-apx. set of min{φ1,φ2}

Convex functions: Wiis Ki-apx. set of φi, (summation of apx.) W1 W2 is a max{K1, K2}-apx. set of φ1+φ2

Useful when access to φis either impossible of very costly

Theorem: efficient succinct approximation of: • a general φ via a unimodal apx. oracle with given argmin • a monotone φ via a general apx. oracle • a convex φ via a general apx. oracle that maintains the structure ofφ.

Example:Let f(x, y) = (x – 2y). It is convex over R2

g1(x) =min yRf(x, y) is convex over R

g2(x) =min yZf(x, y) is NOT convex over R!

- The knapsack problem
- Motivation for stochastic and oracle settings
- Approximating functions in logarithmic space and time
- Applications in the design and analysis of approximation algorithms

g5

z6

g3

g1

g4

g2

z3

z4

z2

z1

z5

Example:5 periods DP zi(I)=minx{gi (I,x)+zi+1(fi(I,x))} (*)

Theorem: stochastic monotone/convex DP admits an FPTAS

Proof: Recursively apply DP equation T times with apx. functions and apx. sets with K=1+ /2T. By the inequality (1+x/n)n<1+2x we get that KT<1+

Theorem [H+09]: Let gt*, z*t+1 be L1,L2-apx. functions of gt, zt+1. Then zt*(x):=minxt Wx,1Wx,2{g*t (It, xt) +z*t+1(ft(It, xt))}is a (general) max{L1,L2,min{K1L1, K2L2}-apx of zt(It)

Theorem: efficient succinct apx. of monotone/convex z via general apx. z* that maintains the structure of z.

• Modular framework for deriving FPTAss

• Functional point of view

• Using primal DP formulation

• Propagation of error via Calculus of K-apx. functions

• Compactify the action space via K-apx. sets

• Speed up construction of K-apx. sets via Calculus of K-apx. sets

• generate new rules for the calculi (“open code” approach) • specialize the calculi to new classes of functions • instead of using exact oracles use FPTASs for them • develop other recursive structures to plug into the framework [HLS09]

References

[DGV04]= B.C. Dean, M.X. Goemans, and J. Vondrak[Ho95] = D.S. Hochbaum[IK75] = O.H. Ibara and C.E. Kim [KN09] = S. Kameshwaran and Y. Narahari[Wo00] = G.J. Woeginger