When Does a Dynamic Programming Formulation Guarantee the Existence of a Fully Polynomial Time Approximation Scheme (FPTAS)?

Abstract

We derive results of the following flavor: If a combinatorial optimization problem can be formulated via a dynamic program of a certain structure and if the involved cost and transition functions satisfy certain arithmetical and structural conditions, then the optimization problem automatically possesses a fully polynomial time approximation scheme (FPTAS). Our characterizations provide a natural and uniform approach to fully polynomial time approximation schemes. We illustrate their strength and generality by deducing from them the existence of FPTASs for a multitude of scheduling problems. Many known approximability results follow as corollaries from our main result.

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

“…When the action spaces are "large" this number of calls can be exponential in the input size. In [32], Woeginger designs a framework for deriving an FPTAS for deterministic DP. Among other assumptions, he requires the cardinality of the action space to be bounded by a polynomial over the binary input size (Condition C.4(ii) in [32]).…”

“…While most of these works are not constructive, the one of Woeginger [32] gives sufficient conditions for a deterministic dynamic program to admit an FPTAS, and states a clear FPTAS for such a DP. He demonstrates a number of examples that fit into his framework, where all of those examples have already (other) known FPTASs.…”

“…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]. In their paper they question whether their model captures other dynamic programming algorithms.…”

“…Early work on FPTASs was pioneered by Horowitz and Sahni [18], Ibarra and Kim [19], and Sahni [28], and since then, the most common techniques for constructing FPTASs are dominance (i.e., omitting states of the DP and actions which are dominated, or approximately dominated, by another state or action) and scaling and rounding the data (see, for example, Section 2.5 in [3] and [17]). Woeginger [32] made a key observation that many FPTASs were designed by modifying DPs, and he designed a framework for deriving FPTASs for deterministic dynamic programs satisfying certain regularity conditions. His framework encompassed results from a dozen of optimization problems, including the knapsack problem, for which the first FP-TAS was developed in the seminal work of Ibarra and Kim [19].…”

“…His framework encompassed results from a dozen of optimization problems, including the knapsack problem, for which the first FP-TAS was developed in the seminal work of Ibarra and Kim [19]. At the same time, [32] did not address a number of deterministic problems that were known to have FPTASs, including treelike variants of the knapsack problem, problems involving convex or monotone functions, and stochastic optimization problems. Many FPTASs are easily constructed once the key ideas from [19,28] are understood.…”