I agree with Consultme. Your notice period counts from the date of acceptance of resignation. If your HR has not accepted your resignation, you cannot claim that they are not relieving you. Of course, no company has the right to hold an employee from opting for the employer of their choice except under legal, fair, and just agreements, if any.
Before accepting the conditions of a new employer, you should have cleared it with your present company, and only after that, agree on the date of joining the new company.
Anyways, now all you can do is request and pursue with the existing company.
Wish you good luck!
Shailesh Parikh
[Phone Number Removed For Privacy-Reasons]
Vadodara