# Minimization of _n_th Order Rate Matching in Satellite Networks with One to Many Pairings
## 1 Introduction
### Our Contribution  
Generally, the recent literature has expressed interest in the concept of appropriate GW-UE pairings in HTS systems to optimize different performance metrics [6; 9; 10]. Particularly, suboptimal one-to-many pairings are provided for the minimization of both losses and rate matching (RM) in [9], in which a special case with \(n=1\) for the latter is provided, while here, the \(n\)th order rate matching is examined. The one-to-many pairings, in the previous paper and the current paper, are investigated to model the important scenario where less GWs have to serve more UEs in realistic scenarios of HTS systems. The rate matching is a widely known performance metric in satellite networks [6; 9; 24]; however, to the best of our knowledge, the influence of the \(n\)th power in the pairings originated by the minimization of the rate-matching function has not yet been investigated. In the current work, this case is studied, and useful conclusions are extracted. Suboptimal one-to-many GW-UE matching, based on the gateways' offered and users' requested capacities, namely OCs and RCs, similarly to this work, is investigated in [10] where the maximization of both the minimum and total system's satisfaction ratio and minimization of both the maximum and total system's dissatisfaction ratio are studied. Due to the difficulty of the aforementioned binary optimization problems, in all cases, a two-step heuristic approach is shown. Firstly, the optimization problem from the binary case is \"relaxed\", and a continuous pairing solution in [0,1] is extracted. Hence, an iterative algorithm is implemented, and then, a continuous to binary feasible transformation of this solution is applied. Specifically, in the related literature, [9; 10], in metrics of losses, rate matching and dissatisfaction ratio, that include quadratic functions, the continuous pairings are extracted by the convex-concave procedure (CCP) [25]. This is a known method for the solution of difference of convex (DC) optimization problems. The CCP algorithm theoretically converges to a stationary point of the optimization problem. The aforementioned two-step approach is also implemented here for the minimization of the \(n\)th order rate matching. The main contributions of the current work, based on GWs' OCs and UEs' RCs, are summarized below:  
* The problem of optimal one-to-many (O2M) pairs extracted by minimization of the \(n\)th order rate matching is a difficult non-convex problem, and the optimal solution has exponential-time complexity. Thus, a fast convergence mechanism is presented to address this problem for the first time to the best of our knowledge. To do that, the initial problem is \"relaxed\" and after appropriate transformations, quadratic forms appear. Then, by using the binomial expansion (BE) and considering positive integer \(n\), to guarantee the BE convergence ([26], Equation (5.12)), we prove that BE includes convex and concave terms. Afterwards, the CCP method is directly applied to solve the problem, and an iterative scheme with low complexity is presented. The proposed two-step approach can be used as a benchmark compared to other algorithms for facing similar problems in the future. * Assuming even or odd \(n\), two different problems are solved. The solution of both is based on the CCP algorithm whose outcome depends on the initial feasible points [9; 10], because non-convex functions, as in our case, have multiple stationary points. The relative error among the rate matching originated by the proposed scheme and the corresponding from exhaustive mechanism, exploring all the feasible pairs, becomes generally greater as \(n\) increases. This can be explained by the fact that in larger \(n\), the binomial expansion includes more factors, resulting in more linear approximations by the CCP approach, ending up with lower performance. However, for smaller \(n\), the performance is ameliorated.