By " forgive", do you mean continue the relationship with that person?
I think in most cases it will not work.
However, I was able to re-establish my relationship after infidelity. It took a lot of work and a lot of time. The person who lied and had an affair has to do a lot of growing up and a lot of personal work to prove him/herself again.
I forgive, but I dont forget. People who lie frequently repeat.

It is a board question with many aspects not addressed. Kind of hard to answer such a question, but here goes...

First, it does depend on the details of the lie / cheat. Details do matter.
Second, it also depends on the person who lied / cheated. Who they are and will / can be...
Third, it also depends on what the forgiver wants in forgiving such a person who has lied / cheated. Keeping in mind that forgiveness is not done for those who has done wrong, but for the healing of those who have been hurt but such wrong doing.

Best I got without knowing more.

I agree, it's a very broad question. A pattern seldom changes, an error in judgment may be a whole different answer. Some people may even see it as throwing stones in a glass house. People are capable of change, but your feelings may not.

A liar implies they constantly lie, hence they most likely won't change. If they lied once, then that could be forgiven and it's not necessarily a personality trait.
Cheating can also be in different categories... Dating, committed relationship, married, age. In my personal opinion, if the person lied to cover up their cheating and are still not being completely honest, they'll do it again.