# Minimizing Interference-to-Signal Ratios in Multi-Cell Telecommunication Networks
## 5 Conclusions  
We proposed a robust and deterministic algorithm to solve the total interference minimization problem. We demonstrated through analysis and simulation that it provides higher quality solutions than the popular Spectral Clustering method (quality as measured by the tinf value of the solution). The algorithm runs quickly enough to be considered in real-world applications. The results of the different simulations plotted on the figures in the previous section demonstrate that our heuristic DP-Similarity Clustering algorithm provides high-quality solutions with low time complexity for the total interference minimization problem in bipartite graphs. Applying it for typical wireless networks, it nicely optimizes the total interference in the overall communication network. Compared with the state-of-the-art spectral clustering method, it is clear that the proposed algorithm achieves better performance with much less (computational) complexity. Figure 6: Scenario 2: A demonstration of the output of the two clustering algorithms on 50 BSs and 200 users. Clustering solutions shown for \(M=40\). Triangles and circles represent BSs and users, respectively. The difference between the solutions produced by similarity clustering and spectral clustering are subtle: one-one larger cluster from each solution is subdivided into two-two smaller clusters by the other solution. (a) A total of 50 BSs and 200 users, randomly distributed according to Scenario 2 (users with zero signal strength to any base station are not shown). (b) DP-Similarity Clustering into \(M=40\) clusters, Algorithm 2; \(\texttt{tinf}\approx 0.438\). (c) Spectral clustering into \(M=40\) clusters, see [3]; \(\texttt{tinf}\approx 0.406\). Next, we list two suggestions for future research problems, whose solutions can considerably increase the usefulness of our algorithm in the wireless network domain. The first problem is this: suppose that the clustering problem is restricted so that any cluster may contain at most \(T\) BSs; assume also that we want to have (at most) \(M\) clusters in the solution. However, prescribing both the upper bounds \(M\) and \(T\) may prevent the existence of feasible solutions, for example, if the number of base stations is more than \(M\times T\). It is easy to construct examples where the greedy DPH-clustering algorithm (Algorithm 2) is not able to satisfy the two conditions simultaneously, even though many feasible solutions exist. It is very probable that some form of backtracking capability could help tremendously, but we have not tried to address this problem yet. A more particular problem can be described as follows: our proposed algorithm assigns every BS to some cluster in the total interference minimization problem. However, it is possible that using a certain BS in any cluster causes more interference than not using it at all. This problem can be dealt with a trivial post-processing procedure: after the clustering \(\mathcal{P}=\{P_{1},\ldots,P_{M}\}\) is determined, delete base station \(i\) from \(P_{k}\) if the removal decreases \(\widehat{w}(P_{k})/w(P_{k})\), since removing a BS from a cluster cannot increase the interference fractions of other clusters. The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/a16070341/s1](https://www.mdpi.com/article/10.3390/a16070341/s1)]([https://www.mdpi.com/article/10.3390/a16070341/s1](https://www.mdpi.com/article/10.3390/a16070341/s1)). Conceptualization, P.L.E. ; methodology, P.L.E. and T.R.M. ; software, T.R.M. ; formal analysis, T.R.M. ; writing--original draft preparation, P.L.E. and T.R.M. ; writing--review and editing, T.R.M. ; visualization, T.R.M. ; project administration, P.L.E. ; funding acquisition, P.L.E. and T.R.M. All authors have read and agreed to the published version of the manuscript. The authors were supported in part by the National Research, Development and Innovation Office--NSFIH grant SNN 135643, K 132696. Data Availability Statement: Data is contained within the article or supplementary material.