The online bipartite matching problem, extensively studied in the literature, deals with the allocation of online arriving vertices (items) to a predetermined set of offline vertices (agents). However, little attention has been given to the concept of class fairness, where agents are categorized into different classes, and the matching algorithm must ensure equitable distribution across these classes. We here focus on randomized algorithms for the fair matching of indivisible items, subject to various definitions of fairness. Our main contribution is the first (randomized) non-wasteful algorithm that simultaneously achieves a $1/2$ approximation to class envy-freeness (CEF) while simultaneously ensuring an equivalent approximation to the class proportionality (CPROP) and utilitarian social welfare (USW) objectives. We supplement this result by demonstrating that no non-wasteful algorithm can achieve an $\alpha$-CEF guarantee for $\alpha > 0.761$. In a similar vein, we provide a novel input instance for deterministic divisible matching that demonstrates a nearly tight CEF approximation. Lastly, we define the ``price of fairness," which represents the trade-off between optimal and fair matching. We demonstrate that increasing the level of fairness in the approximation of the solution leads to a decrease in the objective of maximizing USW, following an inverse proportionality relationship.

Recently, some studies on the fair allocation of indivisible goods notice a connection between a purely combinatorial problem called the Rainbow Cycle problem and a fairness notion known as EFX: assuming that the rainbow cycle number for parameter d (i.e. R(d)) is O(d^β .log(d)^γ), we can find a (1 − ϵ)-EFX allocation with O_ϵ(n^(β/β+1) .log(n)^(γ/β+1)) number of discarded goods. The best upper bound on R(d) is improved in a series of works to O(d^4), O(d^(2+o(1))), and finally to O(d^2). Also, via a simple observation, we have R(d) ∈ Ω(d). In this paper, we introduce another problem in extremal combinatorics. For a parameter l, we define the rainbow path degree and denote it by H(l). We show that any lower bound on H(l) yields an upper bound on R(d). Next, we prove that H(l) ∈ Ω(l^2 / log(l)) which yields an almost tight upper bound of R(d) ∈ Ω(d.log(d)). This, in turn, proves the existence of (1−ϵ)-EFX allocation with O_ϵ(√n .log(n)) number of discarded goods. In addition, for the special case of the Rainbow Cycle problem that the edges in each part form a permutation, we improve the upper bound to R(d) ≤ 2d−4. We leverage H(l) to achieve this bound. Our conjecture is that the exact value of H(l) is ⌊l^2/2⌋ − 1. We provide some experiments that support this conjecture. Assuming this conjecture is correct, we have R(d) ∈ θ(d).