## 5 Properties Analysis of the Functions \(\varphi_{m}^{s}\)
### Single-Resource-Block Maritime User Allocation
The profit of each project \(pro_{I}^{s}\) refers to the WAR value of item \(l\) on resource block \(s\). Moreover, the weight \(wei_{I}^{s}\) represents the power allocated to item \(l\). The knapsack problem stipulates that only one item from each category is selected into the backpack to maximize the total profits. At the same time, the weights are constrained by the \(P_{max}^{s}\) of each category and the maximum capacity \(P_{\text{max}}\) of the backpack. Therefore, when the item \(j\) on resource block \(s\) is allocated to the knapsack, binary variable \(z_{l}^{s}=1\). Finally, \(W_{\text{MCKP}}\) is able to approach the continuous solution of GRAD with an arbitrary accuracy by adjusting \(p\). The error between the optimal result \(W_{\text{MCKP}}*\) of the discrete scheme and the optimal result \(W_{MRB}*\) of the continuous algorithm is as follows:

\[X[a,\,n,\,m]=\widetilde{p}^{*}, \tag{41}\]4

where \(P_{s}^{*}\) refers to the optimal power budget on resource block \(s\) in problem \(W_{MRB}\). The discretization error depends on the upper limit of a linear function in \(\delta\) with the coefficient decided by the system parameters. Appendix B gives the detailed derivation process of Equation (63). \(W_{\text{MCKP}}\) in polynomial time can be dealt with by DP, which is summarized as weighted DP and profit DP [28]. Since the MCKP is able to dynamically solve the maximum profit by discretizing weights, we adopt the principle of weight-based DP. Let \(Z\) be a two-dimensional array. \(Z[s,j]\) is defined as the optimal value of \(W_{\text{MCKP}}\) limited to the first \(s\) categories with a capacity of \(j\times p\). The recursive relationship for \(Z\) on resource block \(s\) is as follows:

\[X[a,\,n,\,m]=\widetilde{p}^{*}, \tag{41}\]5

As shown in Algorithm 6, through the application of the weight-based DP idea to the multiple-choice knapsack algorithm, we propose the use of the MCKP-DP scheme to handle \(W_{\text{MCKP}}\). First of all, we convert the continuous joint optimization problem \(W_{RB}\) to the discrete search space problem \(W_{\text{MCKP}}\). Locating lines 6 and 7, the RBPUA algorithm calculates the profit \(pro_{l}^{s}\) for each project. In line 8, we perform weight-based DP. ### Fully Polynomial Time Approximation Program Based on Dynamic Programming (DP-FPTA)

Aiming at an NP-hard optimization conundrum, the knapsack algorithm and related approximation schemes can effectively ensure that the approximation precision reaches any degree required. Let \(\Pi\) be an NP-hard optimization problem with objective function \(f_{\Pi}\); we propose an approximation scheme for the optimization problem on the input (\(I\), \(\varepsilon\)), where \(I\) represents the instance of \(\Pi\) and \(\varepsilon>0\) is taken as the error parameter. The optimal result of function \(f_{\Pi}(I\), \(\varepsilon)\) satisfies the requirements of Equation (65):

\[\left\{ \begin{array}{l}f_{\Pi}\left(I,\varepsilon\right)\leq(1+\varepsilon) \cdot Optimal,\left(\prod\text{ is a minimization problem}\right)\\ f_{\Pi}\left(I,\varepsilon\right)\geq(1-\varepsilon)\cdot Optimal,\left( \prod\text{ is a maximization problem}\right)\end{array} \right.. \tag{65}\]

This scheme is called the polynomial-time approximation (PTA) algorithm. For each fixed \(\varepsilon>0\), its running time is limited by a polynomial with the size of instance \(I\). In the subsequent more rigorous approximation concept, FPTA limits the running time required by the polynomial with both instance size and \(1/\varepsilon\). From the technical perspective, FPTA is the most providential strategy to solve the NP-hard optimization problem. To deal with the pseudo-polynomial complexity of \(J\), we will formalize this concept and develop a complete polynomial approximation algorithm by combining it with dynamic programming. The core of the DP-FPTA algorithm is that for any \(\varepsilon>0\), the final optimization result is higher than the lower bound of performance guarantee: \((1-\varepsilon)\cdot W_{MKCP}*\).