Replication is a widely used technique in information retrieval and database systems for providing fault-tolerance and reducing parallelization and processing costs. Combinatorial models based on hypergraph partitioning are proposed for various problems arising in information retrieval and database systems. We consider the possibility of using vertex replication to improve the quality of hypergraph partitioning. In this study, we focus on the Balance Preserving Min-Cut Replication Set (BPMCRS) problem, where we are initially given a maximum replication capacity and a K-way hypergraph partition with an initial imbalance ratio. The objective in the BPMCRS problem is finding optimal vertex replication sets for each part of the given partition such that the initial cutsize of the partition is improved as much as possible and the initial imbalance is either preserved or reduced under the given replication capacity constraint. In order to address the BPMCRS problem, we propose a model based on a unique blend of coarsening and integer linear programming (ILP) schemes. This coarsening algorithm is based on the Dulmage-Mendelsohn decomposition. Experiments show that the ILP formulation coupled with the Dulmage-Mendelsohn decomposition-based coarsening provides high quality results in feasible execution times for reducing the cost of a given K-way hypergraph partition.