This paper studies the fair distribution problem of indivisible items, and provides new insights into the EFX problem, which is considered a central unsolved problem in the field of fair distribution, and the PMMS problem, a more rigorous variant of EFX. First, we establish a formal separation between EFX and PMMS by showing that PMMS allocation does not exist for three agent instances with two monotonic cost functions and one additive cost function. Then, we prove the existence of fair allocation for three important special cases: personalized bivalued valuations, PMMS allocation when $a_i$ is divisible by $b_i$, and binary-valued MMS-feasible valuations. Finally, we show that PMMS allocation exists for pair-demand valuations, which are extensions of single-demand cost functions to at most two items. We provide polynomial-time algorithms for all existence results.
