Two-dimensional switching lattices including four-terminal switches are introduced as alternative structures to realize logic functions, aiming to outperform the designs consisting of one-dimensional two-terminal switches. Exact and approximate algorithms have been proposed for the problem of finding a switching lattice which implements a given logic function and has the minimum size, i.e., a minimum number of switches. In this article, we present an approximate algorithm, called JANUS, that explores the search space in a dichotomic search manner. It iteratively checks if the target function can be realized using a given lattice candidate, which is formalized as a satisfiability (SAT) problem. As the lattice size and the number of literals and products in the given target function increase, the size of a SAT problem grows dramatically, increasing the run-time of a SAT solver. To handle the instances that JANUS cannot cope with, we introduce a divide and conquer method called MEDEA. It partitions the target function into smaller sub-functions, finds the realizations of these sub-functions on switching lattices using JANUS, and explores alternative realizations of these subfunctions which may reduce the size of the final lattice. Moreover, we describe the realization of multiple functions in a single lattice. Experimental results show that JANUS can find better solutions than the existing approximate algorithms, even than the exact algorithm which cannot determine a minimum solution in a given time limit. On the other hand, MEDEA can find better solutions on relatively large size instances using a little computational effort when compared to the previously proposed algorithms. Moreover, on instances that the existing methods cannot handle, MEDEA can easily find a solution which is significantly better than the available solutions.