Most research involves incrementally improving existing ideas and methods. Beware of working on problems that are very well studied. I call this practice "trying to build a better mousetrap." It is unlikely that you could build a better mousetrap. Many people have already worked on that problem. Even if you are creative and diligent, your efforts would probably end in disappointment.
Remember that researchers only publish their successes and not their failures. After studying the literature on a subject, you might think "No one has ever tried this approach." However, it is quite likely that others have tried that approach. They tried and failed and did not publish.
I would not wish to dissuade you from trying to build a better mousetrap, just be aware of the realities of working on a well-studied problem. Have a plan for getting the most insight with the least amount of effort. Invest your time wisely. Make sure that you get the state of the art software with which to compare your new method. A poor implementation of an optimal algorithm may have poor performance. Don't assume that an implementation is efficient simply because it is popular. Finally, you may be able to make some headway if you are limiting your focus to special cases.