  We consider the problem of fairly allocating a combination of divisible and
indivisible goods. While fairness criteria like envy-freeness (EF) and
proportionality (PROP) can always be achieved for divisible goods, only their
relaxed versions, such as the ''up to one'' relaxations EF1 and PROP1, can be
satisfied when the goods are indivisible. The ''up to one'' relaxations require
the fairness conditions to be satisfied provided that one good can be
completely eliminated or added in the comparison. In this work, we bridge the
gap between the two extremes and propose ''up to a fraction'' relaxations for
the allocation of mixed divisible and indivisible goods. The fraction is
determined based on the proportion of indivisible goods, which we call the
indivisibility ratio. The new concepts also introduce asymmetric conditions
that are customized for individuals with varying indivisibility ratios. We
provide both upper and lower bounds on the fractions of the modified item in
order to satisfy the fairness criterion. Our results are tight up to a constant
for EF and asymptotically tight for PROP.
