The project focuses on answering the question "How to divide a collection of objects into two groups such that the objects within each group can interact while the objects from different groups rarely interact?" As is well known, the past two decades have witnessed an exponential increase in the use of social or other networks, which consist of objects and connections between them, and carry a lot of information. These networks can be used to detect different groups of objects, each with similar characteristics or preferences. In materials science, advanced engineering alloys, such as steels and high-entropy alloys, may be thought of as 3-dimensional graphs with atoms residing at the nodes of a graph and its edges as bonds. These materials are often polycrystalline, and sudden, unexpected failures occur frequently in the form of fractures along crystal boundaries. Such failures are extremely costly, resulting in, for example, oil and gas spills and even bridge collapses. All the above problems can be cast as balanced graph cut problems, which are very challenging. In the literature, many existing approaches resort to approximate solutions. However, these solutions may differ significantly from the optimal ones. This proposal mainly focuses on the development of efficient and reliable methods for finding optimal graph cuts, which could significantly promote their applications to practical problems in the fields of materials science and social sciences. Partitioning a larg