Toward a Model for Backtracking and Dynamic Programming

Alekhnovich, Michael; Borodin, Allan; Buresh-Oppenheim, Joshua; Impagliazzo, Russell; Magen, Avner; Pitassi, Toniann

doi:10.1109/ccc.2005.32

Cited by 22 publications

(62 citation statements)

References 31 publications

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“…First, Alekhnovich et al present a model for backtracking and DP [2]. They prove several upper and lower bounds on the capabilities of algorithms in their model and show that their model captures the DP framework of [79].…”

Section: Approximating Profitsmentioning

confidence: 99%

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Fully Polynomial Time Approximation Schemes for Stochastic Dynamic Programs

Halman¹,

Klabjan

Li³

et al. 2014

SIAM J. Discrete Math.

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Abstract. We present a framework for obtaining fully polynomial time approximation schemes (FPTASs) for stochastic univariate dynamic programs with either convex or monotone single-period cost functions. This framework is developed through the establishment of two sets of computational rules, namely, the calculus of K-approximation functions and the calculus of K-approximation sets. Using our framework, we provide the first FPTASs for several NP-hard problems in various fields of research such as knapsack models, logistics, operations management, economics, and mathematical finance. Extensions of our framework via the use of the newly established computational rules are also discussed.Key words. fully polynomial time approximation schemes, stochastic dynamic programming, K-approximation AMS subject classifications. 68Q25, 68W25, 90B05, 90B06, 90C15, 90C39, 90C40, 90C56, 90C59

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Section: Approximating Profitsmentioning

confidence: 99%

“…Clearly, ϕ is convex over R 2 . Define ψ 1 , ψ 2 : R→R such that ψ 1 (x) = min y∈R ϕ(x, y) and (2.1) ψ 2 (x) = min y∈Z ϕ(x, y).…”

Section: Notation For Convex Dpsmentioning

confidence: 99%

Fully Polynomial Time Approximation Schemes for Stochastic Dynamic Programs

Halman¹,

Klabjan

Li³

et al. 2014

SIAM J. Discrete Math.

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“…We claim that almost all greedy algorithms 2 utilize such IIA orderings. 2 One example of a non-IIA ordering is the randomized greedy algorithm for the stochastic knapsack problem in [18].…”

Section: Deterministic Vs Randomized Priority Algorithms and The Minmentioning

confidence: 99%

Randomized Priority Algorithms

Angelopoulos

2004

Approximation and Online Algorithms

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Borodin, Nielsen and Rackoff [13] introduced the class of priority algorithms as a framework for modeling deterministic greedy-like algorithms. In this paper we address the effect of randomization in greedy-like algorithms. More specifically, we consider approximation ratios within the context of randomized priority algorithms. As case studies, we prove inapproximation results for two well-studied optimization problems, namely facility location and makespan scheduling.

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“…First, Alekhnovich et al [2] present a model for backtracking and dynamic programming. They prove several upper and lower bounds on the capabilities of algorithms in their model, and show that it captures the simple dynamic programming framework of Woeginger [32].…”

Section: Theorem 51 (Fptas For Monotone Dp) Every Monotone Dynamic mentioning

confidence: 99%

A Fully Polynomial-Time Approximation Scheme for Single-Item Stochastic Inventory Control with Discrete Demand

et al. 2009

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We develop a framework for obtaining (deterministic) Fully Polynomial Time Approximation Schemes (FPTASs) for stochastic univariate dynamic programs with either convex or monotone single-period cost functions. Using our framework, we give the first FPTASs for several NP-hard problems in various fields of research such as knapsack-related problems, logistics, operations management, economics, and mathematical finance. IntroductionDynamic Programming (DP). Dynamic Programming is an algorithmic technique used for solving sequential, or multi-stage, decision problems and is a fundamental tool in combinatorial optimization (e.g., [17], Section 2.5 in [3], and Chapter 8 in [30]). A discrete time finite time horizon dynamic program is to find an optimal policy over a finite time horizon that minimizes the average cost. At the beginning of a time period, the state of the system is observed and an action is taken. Based on exogenous stochastic information, the state, and the action, the system incurs a single-period cost and transitions into a new state. The goal is to find a policy that realizes the minimal total expected cost over the entire time horizon.We can formally model this by means of Bellman's optimality equation. Let z t (I t ) be the cost-to-go (also known as the value function). The value z t (I t ) is simply the cost of an optimal policy from time period t to the end of the time horizon, given that at the beginning of time period t the state is I t . The equation reads (1.1) z t (I t ) = min x t ∈A t (I t ) E Dt {g t (I t , x t , D t ) + z t+1 (f t (I t , x t , D t ))}.

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Toward a Model for Backtracking and Dynamic Programming

Cited by 22 publications

References 31 publications

Fully Polynomial Time Approximation Schemes for Stochastic Dynamic Programs

Fully Polynomial Time Approximation Schemes for Stochastic Dynamic Programs

Randomized Priority Algorithms

A Fully Polynomial-Time Approximation Scheme for Single-Item Stochastic Inventory Control with Discrete Demand

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