## 1 Introduction  
The integrated sensing and communication (ISAC) framework has been proposed as one of the critical technologies in the six-generation (6G) networks, enabling many emerging applications such as virtual reality, smart city, autonomous driving, etc. [1]. The application scenarios mentioned above require a high data transmission rate while ensuring target sensing performance. In recent works [2, 3, 4, 5], a tight combination of sensing and communication functions in ISAC systems has been achieved through a series of designs, including integrated architecture, waveforms designing, etc. By achieving the sharing of spectrum and wireless infrastructure, the ISAC technology improves resource efficiency and utilization, and reduces signal interference and hardware overhead [6].  
However, despite the enormous benefits of ISAC technology, its applications face considerable challenges in practice due to the obstacles of dense buildings or landscape trees in urban environments [7]. Unlike communication systems in which both line-of-sight(LoS) and non-LoS (NLoS) links can be leveraged for data transmission, the radar sensing function relies on the LoS link between the transmitter and the target area, while the NLoS link is considered to be an interference [8]. Therefore, exploring the target sensing problem for the ISAC system without an LoS link is necessary [9].  
The intelligent reflecting surface (IRS) is a promising technology in next-generation wireless systems due to its excellent ability to reconstruct wireless environments [10]. By manipulating the phase shifts and amplitude of reflecting elements, the IRS creates the virtual LoS link in NLoS areas. Motivated by the advantages of IRS in reconstructing the wireless propagation environment, it is natural to exploit IRS to assist ISAC systems to improve communication data rate and enhance sensing accuracy and resolution [11]. In the IRS-assisted ISAC system, multiple beams can be synthesized for the user and the desired signal can be enhanced by the joint design of phase shift and transmit beamforming [9; 12]. Moreover, the IRS reduces hardware and energy overhead using low-cost passive components without needing a radio frequency (RF) unit [13]. Hence, high spectrum efficiency and low cost advantages prompt us to research IRS-assisted ISAC systems.  
Although the IRS-assisted ISAC system shows significant potential, its implementation still faces challenges, such as the joint design of phase shift and beamforming matrices. The ISAC system's data calculation and signal processing are generally complex and require more resources. Due to the constrained computation and energy resources of the user terminal, the heavy sensory data processing load of user equipment (UE) is solved by mobile edge computing (MEC) technology. MEC works by offloading the computational task from UE to the edge network and achieving better time efficiency and performance [14]. This work investigates the joint resource allocation and task offloading optimization problem in the multi-user IRS-assisted ISAC scenario. In particular, power and spectrum resources are allocated by beamforming and phase shift design, while computing resources are allocated by task offloading.