Licensed spectrum of Long Term Evolution (LTE) is being strained with the increased traffic demand. To cope with this problem, mobile operators use the unlicensed spectrum to complement the available licensed spectrum. However, using LTE in unlicensed spectrum introduces new problems, such as the coexistence between LTE and other unlicensed communication technologies (e.g., Wi-Fi). This paper proposes a new coexistence mechanism to provide a fair coexistence between LTE and Wi-Fi networks on unlicensed bands. The proposed mechanism assigns resource blocks in eNodeB to LTE users in such a way that it provides efficient and fair usage of unlicensed spectrum for all LTE users and the Wi-Fi networks. We formulate the wireless resource allocation and fair coexistence problem between LTE and Wi-Fi as a non-linear integer programming. Then, we propose an approximation algorithm, which is a Polynomial-Time Approximation Scheme (PTAS), to solve this problem. Our numerical results validate the proposed mechanism's performance vis-a-vis other existing solutions.

Clustering is an effective method for improving the network lifetime and the overall scalability of a wireless sensor network. The problem of balancing the load of the cluster heads is called load‐balanced clustering problem (LBCP), which is an NP‐hard problem. In this study, the authors use parameterised complexity to cope with this NP‐hard problem. The authors show that LBCP can be solved by a k ‐additive approximation algorithm with a running time of , where k is an upper bound on the maximum load assigned to the cluster heads and n is the input size. Also, LBCP is FPT with respect to the maximum load of the sensor nodes and the number of sensor nodes. The authors propose an fpt‐algorithm with respect to these parameters for this problem. In addition, they prove that LBCP is when the number of the cluster heads is selected as the parameter.