1 / 22 sublinear fptass for stochastic optimization problems nir halman, huji based on joint works...
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Sublinear FPTASs for Stochastic
Optimization Problems
Nir Halman, HUJI
Based on joint works with D. Klabjan, C-L Lee, M. Mostagir, J. Orlin and D. Simchi-Levi
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FPTASs
Def: An FPTAS (Fully Poly. Time Approximation Scheme) is a (1+ ε)-apx. alg. that runs in time poly. in |x| and 1/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
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Talk Outline• 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
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Knapsack Problem (KNP)
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)=?
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
NP-C, admits an FPTAS [IK75] (by scaling a dual DP)
Can be calculated in O(TB) time, i.e., pseudopolinomial in input size
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KNP – oracle setting
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)
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KNP - stochastic settingDef.: 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
2) Benefit of adaptivity [DGV04]…
1) Order of items considered is now important…
3) Fact: Some polynomially-solvable problems stop being such in stochastic setting
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KNP - stochastic ordered settingDef.: 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
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
DP form.: zi(Ii) = maxx=0,…, B/vi EDi{xpi+zi+1(Ii –x(vi+Di))},
(total profit to gain with remaining Ii volume by ai,…,aT) z1(B)=?
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Talk Outline
• 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
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A Question
Def: • φ* is a K-apx. of φ if φ(x) ≤ φ*(x) ≤ Kφ(x), x (note that if φ has 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 is unimodal if x* so φ is decreasing until x* and increasing afterwards
Under what conditions an oracle function φ:D→R+ admits an efficient succinct K-approximation? (assume finite DR and K≥1. Input size = log |D|+log φmax)
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1. K-approximation Sets
Definition [H+08]: Let φ:D→Z+ be unimodal. A K-apx. set of φ is a subset W with argminφ, Dmax, DminW D and the ratio between the values of φ on each two consecutive points in W is at most K
φ
φ*
Construction: φ* is the apx. of φ induced by W if:
W K
K
Answer: Logarithmic in the input size. Moreover, it can be constructed in logarithmic time and # of oracle queries
Question: How small a K-apx. of φ can be?
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K-approximation Sets – Cont’
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 preserves the structure of φ
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2. Calculus of K-apx. FunctionsCalculus of K-apx functions [H+08]: α,β ≥ 0, φi* a Ki-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
Corollary [H+08]:(mimization of summation of composition) Let gt*, z*t+1 be L1,L2-apx. functions of (unimodal) gt, zt+1 then
zt*(I)=minxAt(I){gt*(I, x)+z*t+1(ft (I, x))} is a max{L1,L2}-apx of zt(I)
Optimality equation: zt(I) = min xAt(I) {gt (I, x) + zt+1(ft (I, x))}
Theorem [H+08]: Let Wx,1,Wx,2 be Ki-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)
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Subtraction of approximation
Theorem [HOS11]: φ1* a K1-apx. of φ1, φ2* a K2-apx. of φ2 . If φ2cφ1 for c<1/K1K2, then φ1*-φ2* is a controlled apx. of φ1-φ2
Theorem [HOS11]: Let φi* be an Li-apx. of φ1 and Wi, be a Ki-apx. set of φi
*, i=1,2.
If φ2 cφ1 for c<1/K1L1L2 then 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))}
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3. Approximating CDFsLet F() be a CDF of integral non-negative r.v. D bounded by M, i.e., F(x)=Prob(Dx). Let
where ψ is a monotone non-decreasing step function with break points a1,...,an. We decompose ψ as 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 of n non-decreasing functions (n is poly., M is not)
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4. Calculus of K-approximation Sets
Unimodal functions: α,β ≥ 0,Wi is Ki-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}
Focuses on the domain of the functions
Monotone functions (of the same kind):Wi is 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: Wi is Ki-apx. set of φi, (summation of apx.) W1 W2 is a max{K1, K2}-apx. set of φ1+φ2
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5. Approximated Access to φ
Useful when access to φ is 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 of φ.
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6. Discrete Convexity
Example: Let f (x, y) = (x – 2y). It is convex over R2
g1(x) = min yR f (x, y) is convex over R
g2(x) = min yZ f (x, y) is NOT convex over R!
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Talk Outline
• 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
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Optimization OverFinite Horizon monotone/convex DP’s
Example: 5 periods DP zi(I)=minx{gi (I, x)+zi+1(fi (I, x))} (*)
z6g5g4g3g1 g2 z5z4z3z2z1
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 .
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+
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Our Approach
• 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
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Future Research
• 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]