Regression testing is performed throughout the software lifecycle to uncover the faults as early as possible and to ensure that changes do not have any adverse effect in the software that is operational. Test suites once developed are reused and updated frequently. As the software evolves, test cases in the test suite may become redundant. The reason behind this is that the requirements covered by newly added test cases may also be covered by the existing test cases. This redundant nature of test suite increases the cost of executing the same. Further, resource and time constraints impose the necessity to develop techniques to minimize test suites by removing redundant test cases. Few heuristic approaches have been used to solve the test suite minimization problem. Even though solutions exist, still the redundancy of test case remains. In order to solve this problem, this paper proposes two Harrold-Gupta-Soffa (HGS) based heuristic algorithms namely, Non Redundant HGS and Enhanced HGS. The former utilizes the redundant strategy available with Greedy, Redundant, Essential (GRE) to get rid of redundancy, whereas the latter selects a test case for higher cardinalities based on overall coverage of unmarked associated testing sets and thus arrives at reduced, non-redundant test suite. The experiments show that the proposed algorithms always select smaller size of test suite, compared to the existing HGS heuristics.