## 1 Introduction  
Device-to-device (D2D) communication has been proposed as an underlay to cellular networks where user equipments (UEs) in proximity are able to communicate directly with one another with the support of evolved node B (eNB) [1, 2]. D2D communication pairs bring a large amount of benefits for users, such as expanding the service coverage area, increasing data rate, improving system energy efficiency, enhancing network throughput, reducing network delay, creating new types of services, and so on [3, 4]. Additionally, the spectral efficiency could be further improved by allowing D2D users (DUs) to share the same spectrum resources with cellular users (CUs). In this case, intra-cell interference between DUs and CUs is no longer negligible so that a proper radio resource allocation strategy is needed to obtain the potential performance gains. Most of the existing works, so far, have been carried out on resource allocation for hybrid D2D-cellular systems in order to improve spectrum efficiency and mitigate the interference between D2D and cellular communications. In [5], an interference limited area was defined for uplink underlaying systems, where no CUs can reuse the same spectrum resource with D2D pair to avoid strong interference. Various resource reusing ways are considered in [6], where eNB can allocate either orthogonal or non-orthogonal radio resources to CUs and D2D users. A new spectrum sharing protocol for D2D communication overlaying a cellular network is proposed in [7], which allows the D2D usersto act as a relay to assist the two-way communications between the eNB and the CUs. The authors of [8] developed an optimal and low-complexity channel reusing algorithm to maximize the number of admitted D2D pairs. The research in [9] focuses on resource allocation and interference management issue in D2D communication underlaying cellular networks, where the system sum-rate optimization for D2D and cellular links is considered. Both the works of Yu et al. and Jung et al. [10; 11] consider the joint mode selection and resource allocation problem that aims to improve the sum-rate and the power efficiency. While guaranteeing the quality of service (QoS) of both CUs and D2D pairs, a centralized spectrum allocation scheme has been designed to maximize the system throughput in [12]. In [13], a coalitional game approach is employed to solve the joint mode selection and D2D resource allocation. The aforementioned works generally focus on investigating the resource allocation in the case that one D2D pair could reuse only one cellular resource, even though there are much more CUs in practical networks. Actually, the system performance parameters, such as spectral efficiency, will be further improved if a D2D pair could reuse the resources of multiple CUs. Since more subcarriers are available, the DUs would have more opportunities to explore the multi-channel diversity, thereby making spectrum utilization more efficient. There are some works on resource allocation for multicarrier optimization problem in different systems. In [14; 15], the authors propose a resource allocation algorithm design to iteratively find the optimal solution for maximization of the energy efficiency of orthogonal frequency division multiple access (OFDMA) systems based on the Dinkelbach transformation. In [16], a resource allocation algorithm where D2D links can reuse the resources of more than one CU is proposed, while the QoS is not guaranteed for CUs. The authors of [17] study optimal and suboptimal power allocation methods such that the D2D link can maximally utilize multiple possible cellular resources and meanwhile the QoS of CUs is guaranteed, however, they only considered a simple scenario where there only one D2D pair exists. In this paper, we consider a system where multiple D2D links reuse the uplink resources of multiple CUs. We formulate the problem as a mixed integer non-linear programing (MINLP) problem with the objective to maximize the spectrum efficiency of the heterogeneous network.