A bicycle (any other circular object is valid in its/one's absence)
Solution 1:
The first phrase is grammatically correct. The second phrase should be rewritten as "any other circular object is valid in the absence of one." Your use of "one's" would likely seem strange to most English speakers as the use of one as a pronoun is nearly always in reference to an indefinite person (e.g. "One's experiences shape one's expectations.").