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

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 ))}.

We show that a Simple Stochastic Game (SSG) can be formulated as an LP-type problem. Using this formulation, and the known algorithm of Sharir and Welzl [SW] for LP-type problems, we obtain the first strongly subexponential solution for SSGs (a strongly subexponential algorithm has only been known for binary SSGs [L]). Using known reductions between various games, we achieve the first strongly subexponential solutions for Discounted and Mean Payoff Games. We also give alternative simple proofs for the best known upper bounds for Parity Games and binary SSGs.To the best of our knowledge, the LP-type framework has been used so far only in order to yield linear or close to linear time algorithms for various problems in computational geometry and location theory. Our approach demonstrates the applicability of the LP-type framework in other fields, and for achieving subexponential algorithms. Introduction. Sharir and Welzl [SW] defined a model which generalizesLinear Programming (LP) and called it the LP-type model (see definitions in Section 2.2). An LP-type problem of combinatorial dimension d, where d is independent of the size n of the problem, is called fixed dimensional. Several algorithms that solve LP-type problems in time linear in n are known, such as the ones of Sharir and Welzl [SW] or Kalai [Ka]. The O(n) time algorithm of Clarkson [Cl], which was originally formulated to solve LP, fits the LP-type model as well [CM], [GW1]. By formulating problems as fixeddimensional LP-type problems, and using the LP-type algorithms, one can obtain linear time algorithms to various optimization problems, mainly in computational geometry and location theory, as shown in [A] and [MSW].The algorithms of [Ka] and [SW] run in time subexponential in d. In this paper we use the LP-type framework in order to give the first strongly subexponential solution for Simple Stochastic Games, Discounted Payoff Games and Mean Payoff Games (defined below). To the best of our knowledge, this is the first application of the LP-type framework for solving a problem which is neither in computational geometry nor in location theory. Moreover, it is the first application of variable-dimensional LP-type problems.A Simple Stochastic Game (SSG) is defined on a directed graph with three types of vertices, min, max and average, along with two sink vertices, the 0-sink and the 1-sink.

In the classical p-center problem there is a set V of points (customers) in some metric space, and the objective is to locate p centers (servers), minimizing the maximum distance between a customer and his respective nearest server. In this paper we consider an extension, where each customer is associated with a set of existing depots or distribution stations he can use. The service of a customer consists of the travel of a server to some permissible depot, loading of some package (e.g., a spare part) at the depot, and the delivery of the package to the customer. This model is called the customer one-way problem. In the round-trip version of the problem, the service also includes the travel from the customer to the home base of the server. In both problems the customer cost of the service is a linear function of the distance travelled by the server. The objective is to locate p servers, minimizing the maximum customer cost (weighted distance travelled by the respective server).Since the classical p-center problem is NP-hard, so are the one-way and the round-trip models we study. We present efficient constant factor approximation algorithms for these problems on general networks. Turning to special networks, we prove that the one-way problem is strongly NP-hard even on path networks. We then present polynomial time algorithms for the round-trip problem on general tree networks. We also discuss the single center case, and provide polynomial time algorithms for general networks, tree networks and planar Euclidean and rectilinear metric spaces.