Waiving the Notice Period
Yes, the company can waive off the notice period if it desires to do so. The appointment letter states that there should be an ample/prescribed time limit (notice period) to replace the vacant post and ensure a smooth handover of the task/project handled by the resigned employee, which is an operational need. If the company finds an alternative within the company, is at ease, or is discontinuing the project, it can relieve the employee anytime after accepting the resignation letter.
Lastly, an employee who hands in his/her resignation is considered disloyal to the company from that point onwards. The company may choose to pay for the notice period if they desire to do so, but it is not mandatory.