Parameterized complexity of coverage in multi-interface IoT networks: Pathwidth

The Internet of Things (IoT) has emerged as one of the growing fields in digital technology over the past decade. A primary goal of IoT is to connect physical objects to the Internet to provide various services. Due to the vast number and diversity of these objects, referred to as devices, IoT must tackle both traditional and novel theoretical and practical network problems. Among these, multi-interface problems are well-known and have been extensively studied. This research focuses on one of the newest multi-interface models that fits well within the IoT context. It is known as the \textit{Coverage} in the budget-constrained multi-interface problem, where the budget represents the total amount of energy in the network, and coverage refers to the model's goal of activating all required communications among IoT devices. Since most IoT devices are battery-powered, energy consumption must be considered to extend the network's lifespan. This means selecting the most energy-efficient interface configuration that allows all desired connections to function. To achieve this, both global energy consumption and the local number of active interfaces are limited. Moreover, this model also incentivize devices to turn on the available interfaces to create a more performant network. Finally, this model also takes into account the performance of the networks assigning a profit to devices that activate interfaces and realize connections. This problem can be represented using an undirected graph where each vertex represents a device, and each edge represents a desired connection. Every device is equipped with a set of available interfaces that can be used to facilitate transmission among the devices. The final goal is to activate a subset of the available interfaces that maximize the total profit, while not violating the constraints. This problem has been recognized as \NP-hard, which is why we decided to investigate the decision version from the perspective of fixed-parameter tractability (FPT) theory. FPT is an advanced area of complexity theory that aims to identify the core complexity of a combinatorial problem by incorporating parameters into the time complexity domain. We provide two fixed-parameter tractability results, each describing an FPT algorithm. One algorithm is based on the well-known pathwidth parameter, the number of available interfaces, and the maximum available energy. The other algorithm considers pathwidth, the number of available interfaces, and an upper bound on the optimal profit. Finally, we show that these two algorithms can be applied to the maximization version of the problem.