Constraints depict the dependency relationships between parameters in a software system under test. Because almost all systems are constrained in some way, techniques that adequately cater for constraints have become a crucial factor for adoption, deployment and exploitation of Combinatorial Testing (CT). Currently, despite a variety of different constraint handling techniques available, the relationship between these techniques and the generation algorithms that use them remains unknown, yielding an important gap and pressing concern in the literature of constrained combination testing. In this paper, we present a comparative empirical study to investigate the impact of four common constraint handling techniques on the performance of six representative (greedy and search-based) test suite generation algorithms. The results reveal that the Verify technique implemented with the Minimal Forbidden Tuple (MFT) approach is the fastest, while the Replace technique is promising for producing the smallest constrained covering arrays, especially for algorithms that construct test cases one-at-a-time. The results also show that there is an interplay between effectiveness of the constraint handler and the test suite generation algorithm into which it is developed.