Strongly Polynomial FPTASes for Monotone Dynamic Programs

Abstract

In this paper we introduce a framework for the automatic generation of Strongly Polynomial Fully Polynomial Time Approximation Schemes (SFPTASes) for monotone dynamic programs. While some ad-hoc SFPTASes for specific problems are already known, this is the first framework yielding such SFPTASes. In addition, it is possible to use our algorithm to get efficient (non strongly polynomial) FPTASes. Our results are derived by improving former (non strongly polynomial) FPTASes which were designed via the method of K-approximation sets and functions. We demonstrate our SFPTAS framework on five application problems, namely, 0/1 Knapsack, counting 0/1 Knapsack, Counting $s-t$ paths, Mobile agent routing and Counting n-tuples, for the last problem we get the fastest SFPTAS known to date. In addition, we use our algorithm to get the fastest (non strongly polynomial) FPTASes for the following other three application problems: Stochastic ordered knapsack, Bi-criteria path problem with maximum survival probability and Minimizing the makespan of deteriorating jobs.

Appendices

Appendix

A An Example - Finite-Time Discrete-Time Horizon DPs

Consider a finite-time discrete-time horizon DP, as defined in Bertsekas [3]. The model has two principal features: (i) an underlying discrete time dynamic system, and (ii) a cost function that is additive over time. The system dynamics are of the form

$$\begin{aligned} I_{j+1}=\mathrm{tran}_j(I_j,x_j),\quad j=1,\ldots ,T, \end{aligned}$$

where

• j is the discrete time index,

• $I_j$ is the state of the system, which belongs to the state space $\mathcal {S}_{j}$,

• $x_j$ is the action or decision to be selected in time period j, which belongs to the action space $\mathcal {A}_j(I_j)$,

• T is the number of time periods.

The cost function, denoted by $g_j(I_j,x_j)$, is nonnegative and additive in the sense that the cost incurred in time period j is accumulated over time. Let $I_1$ be the initial state of the system. Then the total cost is

$$\begin{aligned} g_{T+1}(I_{T+1})+\sum _{j=1}^T g_j(I_j,x_j), \end{aligned}$$

where $g_{T+1}(I_{T+1})$ is the terminal cost incurred at the end of the process.

The problem is to determine

$$\begin{aligned} z^*(I_1)=\min _{x_j\in \mathcal {A}_j(I_j)\mid 1\le j\le T }\left\{ g_{T+1}(I_{T+1})+\sum _{j=1}^T g_j(I_j,x_j)\right\} . \end{aligned}$$

(14)

The optimization is over the actions $x_1,\ldots ,x_T$. Here, $x_j$ is selected with the knowledge of the current state $I_j$.

The state space and the action space are one-dimensional. Note that the domain of the single-period cost functions $g_j$ and transition functions $\mathrm{tran}_j$ is $\mathcal {S}_j\otimes \mathcal {A}_j$, where $\mathcal {S}_j\otimes \mathcal {A}_i=\bigcup _{I_j \in \mathcal {S}_j}(\{I_j\}\times \mathcal {A}_j(I_j))$. For every initial state $I_1$, the optimal cost $z^*(I_1)$ of the DP is equal to $z_1(I_1)$, where the function $z_1$ is given by the last step of the following recursion, which proceeds backward from period T to period 1 (Bellman recursion):

$$\begin{aligned} \begin{aligned} z_{T+1}(I_{T+1})&=g_{T+1}(I_{T+1}),\\ z_j(I_j)&=\min _{x_j\in \mathcal {A}_j(I_j)}\left\{ g_j(I_j,x_j)+z_{j+1}(\mathrm{tran}_j(I_j,x_j))\right\} ,\quad j=1,\ldots ,T, \end{aligned} \end{aligned}$$

(15)

where $z_j(I_j)$ is the cost-to-go function whose value is the minimal cost when starting at time period t with state $I_t$ and continuing until the end of the time horizon.

We note that (15) is a special case of DP formulation (4) in the following sense. (i) We reverse the order of recursion so that $t=T+1-j$, (ii) we set $m=1$ and for convenience drop the index i from the double index t, i, (iii) the information given in $A_t(I_t)$ is the description of the action space $\mathcal {A}_{T+1-j}(I_{T+1-j})$, (iv) when considering time period t in (4), instead of using all previously-calculated $\{z_r\}_{r<t}$, we use only $z_{t-1}$, i.e., when considering time period $T+1-t$ in (15), we only use the previously-calculated cost-to-go function $z_{T+2-t}$, and (v) $f_0=g_{T+1}$ and for $1 \le t \le T$ we have (let $j=T+1-t$)

$$\begin{aligned} f_t= \min _{x_{j}\in \mathcal {A}_{j}(I_{j})} \left\{ g_{j}(I_{j},x_{j})+z_{j+1}(\mathrm{tran}_{j}(I_{j},x_{j}))\right\} . \end{aligned}$$

B A Comparison between Conditions 1-4 and Conditions 1-4(i) of [2]

There are three differences between the two sets of conditions:

• Condition 2 of [2] requires boundedness of $\log U_{\mathcal {S}}$ while we do not. We replace this requirement with the boundedness of the sets of change points which is implied by Condition 6.

• Our framework aims to achieve strongly polynomial running time, while [2] is not. Therefore, when requiring boundedness of elements or running times, [2] requires boundedness in the (binary) input size, and we request boundedness in the dimension of the input.

• We add the requirement for monotonicity of ${\bar{z}}_{t,i}$ in Condition 3. If for a DP there exist t, i such that ${\bar{z}}_{t,i}$ is not monotone for t, i, then [2] refer this DP to their indirect case, see Remark 3 therein. As we deal only with their direct case, we have to require monotonicity.

C Statement of Function FPTASApxSet