## 1 Introduction  
With increasing advances in wireless communication systems, electronics and sensing technologies, wireless sensor networks (WSNs) have recently become an important, indispensable platform for a great number of vital surveillance and control applications [1]. A WSN consists of hundreds to thousands of tiny sensor nodes that are equipped with the same hardware structure and software capabilities (such as sensing, data processing, communication, etc.) to perform distributed sensing tasks and to transmit or forward information gathered to the base station (BS). Due to flexible use, a convenient layout and a low cost, WSNs have been widely used in military reconnaissance, environmental monitoring, smart home, industrial control, and other application fields [2; 3; 4; 5]. With much more sensor nodes, WSNs are different from the traditional network in many aspects, including the architecture, addressing the method, the communication mode, the routing structure, etc. With the flat routing structure (i.e., a single-layer planer structure), the routing protocol is presently one of the key technologies in WSNs [5; 6]. In general, as sensor nodes are powered by limited energy, energy conservation is commonly considered the most key challenge to guarantee the connectivity of WSNs and to extend the lifetime of sensor nodes, especially when the sensing field is inaccessible and the battery cannot be replaced [7]. Hence, to improve WSN longevity, routing protocols that aim to optimize power usage should become involved to balance the energy consumption of sensor nodes [4; 8]. Topology control is one of the critical strategies for routing protocols in WSNs to improve network longevity. This is dedicated to periodical restoration of network topology in order to achieve the desired connectivity and ensure sufficient routes [9]. Network topology is the foundation of the better connectivity and the optimal route, while it also contributes to effective routing protocols for controlling the network route. Effective routing protocols help to improve WSN longevity by determining which sensor nodes should participate in the network routing operation and which should not at each transmission round [10]. Towards improving WSN longevity, we pay more attention to topology control to form the optimal network structure. However, we abandon the clustering mechanism and propose a dynamic hierarchical protocol based on combinatorial optimization (DHCO) to establish a feasible routing set for each sensor node. The feasible routing set is equivalent to a solution space consisting of many valid routes. The optimal route is obtained from the solution space by using the maximum-minimum criterion, which can be formulated as a combinatorial optimization problem. For the consideration of balancing the energy consumption of sensor nodes, the essence of the DHCO algorithm is to obtain the optimal route for each sensor node from its corresponding feasible routing set. Differently to state-of-the-art WSN routing algorithms (i.e., low-energy adaptive clustering hierarchy--LEACH, hybrid energy-efficient distributed clustering--HEED, genetic protocol-based self-organizing network clustering--GASONeC, double cost function-based routing--DCFR, etc. ), in selecting the cluster head or the next hop node on the basis of the node energy property or the node distance property, the DHCO algorithm employs a hierarchy-based connection mechanism to construct a hierarchical network structure, and utilizes the combinatorial optimization theory to establish the feasible routing set for each sensor node, taking advantage of the maximum-minimum criterion to obtain their optimal route to the base station; this contributes to balancing the energy consumption of sensor nodes and improving WSN longevity. The remainder of this paper is organized as follows. We present related work on routing protocols to improve WSN longevity in Section 2. Section 3 focuses on the system model of WSNs, including the network model and the sensor energy model.