For efficient use of limited electromagnetic wave resource, the assignment of communication channels to call requests is very important in a cellular network. This task has been formulated as an NP-hard combinatorial optimization problem named the channel assignment problem (CAP). Given a cellular network and a set of call requests, CAP requires to find a channel assignment to the call requests such that three types of interference constraints between channels are not only satisfied, but also the number of channels (channel span) is minimized. This paper presents an iterative search approximation algorithm for CAP, called the Quasi-solution state evolution algorithm for CAP (QCAP). To solve hard CAP instances in reasonable time, QCAP evolutes quasi-solution states where a subset of call requests are assigned channels and no more request can be satisfied without violating the constraint QCAP is composed of three stages. The first stage computes the lower bound on the channel span for a given instance. After the second stage greedily generates an initial quasi-solution state, the third stage evolutes them for a feasible channel assignment by iteratively generating best neighborhoods, with help of the dynamic state jump and the gradual span expansion for global convergence. The performance of QCAP is evaluated through solving benchmark instances in literature, where QCAP always finds the optimum or near-optimum solution in very short time. Our simulation results confirm the extensive search capability and the efficiency of QCAP.